Landing in Debrecen on 1 April (Easter), are there trains to Budapet on this day?

Yes
http://elvira.mav-start.hu/elvira.dll/uf?language=2&i=Debrecen&e=BUDAPEST*&d=2018.04.01&mikor=-1&u=1156&go=

Thank you.
:-)

I'd appreciate help also on the matter of car rental.
Pick up from Budapest on 4 April early AM and return Debrecen (city or airport) on 7 April evening (say 19:00).
We fly out at 06:00 on the 8th so we'll need to return the can the day before.
I tried rentalcars.com and a couple of hours ago they had many options available. But now I receive a message saying that "We're sorry, but the companies we work with in Budapest don't have any cars available."
Highly unlikely that all the cars were rented in the past 2-3 hours.

Any idea what I'm doing wrong, or another suggestion regarding alternative reliable car rental companies there?

I tried Avis, Herz, Europcar through dealers here - they said that since 7 April is aSunday - all rental car offices in Debrecen are closed. Very strange.

Thanks.

I'm not much help on car rentals. Never needed a car in Hungary and never wanted to pay to park it. Sorry.

Google Maps agrees that nearly all car rental offices in Debrecen are regularly closed on both Saturdays and Sundays. But this Hungarian company does have weekend open hours at the airport: https://www.autonom.hu/en.

FYI, I'm not sure where in the car booking process the consolidator search engines check office opening times. Sometimes it seems like a step or two into the process. It may be possible to park the car in a designated location and drop the keys in an "after-hours" mailbox, but then you don't get anyone to sign off your paperwork. If doing that, take photos of the condition in which you left the vehicle, just in case.

In the United States and in Hungary; April 7, 2018 is a Saturday. That should help.

Unless you want the rental for some other reason I suspect you could hire a car and driver to get you back to Debrecen for a the same or less than renting a car for 3 days and paying the drop off fee in Debrecen. I know a driver if that interests you.

Hertz
Debrecen Airport
Mikepercsi ut 0493/24
Debrecen , 4030
Hungary
There is not a vehicle rental office at this location. This may require that the customer be taken to the nearby servicing location. Pick-up and/or drop off should be coordinated with the servicing location.
Phone: +36 30 33 69072
Location Type: Independent Licensee
Hours of Operation: Mo-Fr 0800-1800, Sa 1200-1300 1700-1800, Su 1300-1400 1700-1800

Thank you, Laura, I'll look in that Web site.

Thank you, James. I do find it strange that all car rental agencies are closed on a Saturday in the country's 2nd largest city...

We arrive on April 1, will stay in Budapest until April 4th, pick up the car and then travel around the country until April 7h.
I prefer to drive myself.
We fly out early AM on the 8th so I need to return the car on the 7th.

Apparently you can get a car from Hertz near the airport on Sat and Sun. When you get a quote, post it. Im guessing a sub compact for about 45 euro a day after all the fees, maybe 15 euro for the vignette, and 25 euro for gas and 50 EUR for 2 nights parking in Budapest, maybe a 200 euro drop fee. So, maybe 350 to 450 euro total. I'd be curious how close I came.

OK
I'm working with our Hertz brach to resolve this. They'll contact the Debrecen branch to work out the details.

If worst comes to worst, I'll change plans for the 7th and return the car in Budapest downtown in the afternoon and then take the train back to Debrecen.
If this is the solution then I'll have to skip Eger and surroundings.

[edit} Local Hertz returned and said they have the agreement of the Debrecen station, but at the moment the local guys don't have the April prices, so they'll call me back in a couple of weeks to update me.
There's going to be a ~20 Euro surcharge to return the car at a different location.

I’ve never seen a major car rental agency not have an after hours key drop. We have rented cars in Europe more times than I could count and almost always use the after hours drop.

Same here, Emily. This is all very strange. Hope that this will be resolved.
I'll post here after Hertz return to me with the pricing.

Or, with out a car take the train to Eger and a train to Debrecen from Eger. Fortunately its one of the few town to town trips you can make in Hungary without returning to Budapest.

The plan is to go directly to Budapest, stay there for 2.5 days and then leave by car, as we plan to visit several places in the country between 4-7 April, before we need to return to Debrecen prior to our flight back home.
If we were to make "star" tours (ie stay in one central place as base and do day trips from there) - this would be a good idea. Basically this is what I do whenever I'm in London (use the train to travel to the neighboring towns - Oxford, Cambridge, Winchester, Canterbury etc).
Only lately have I gathered enough courage to drive in the UK (to the Cotswolds, Stonehenge etc).

Just my preference, but 2.5 days in Budapest is sort of shy of a good trip. Four full wake up and go to bed in Budapest days is a good minimum. But i am biased as hell.

From Budapest there are a few places very nearby that you can do as full or half day trips with out a car. The Danube Bend or just a part of it. The full bend trip is a bit much for a day but seeing Esztergom and Szentendre in one day is a good trip by train and bus and water taxi. Vac is a good day trip and even Gyor / Pannonhalma is a very doable day trip; but probably a bit better over night. Tihany or some other part of Balaton can be done in a day, but again, an overnight is much better. Kecskemét also is a decent day trip.

I know, the draw back about day trips is back tracking on the way home. That wouldn't be true with Esztergom and Szentendre, the rest yes. What i do is i generally train to the town or village and then I have someone pick me up and we do something creative on the way home. For instance the Bear park between Vac and Budapest.

Sadly the deal was only 1-8 April... Would love to have more time in Budapest.

Update.
After checking several options, the only applicable one was to rent via Hertz. They work with a local company that allows to take the car in Budapest and return it in Debrecen late on the 7th (via a key drop box).
They do not charge for this but I'll need to pay 45 Euro + tax for the car return in a different city.
So this issue is settled.
Thank you all for your help.

OK
Returned from the trip, here's what happened regarding to the train.
Reserved a taxi ride with Fonix from Debrecen airport to the train station.
We landed 30 mins late due to a late takeoff.
The taxi obviously did not wait for us. Tried to reach the office via phone and email, got no response.
So we went outside (very cold, strong winds and drizzle) and were able to catch a taxi within minutes from the parking lot right outside the terminal. It took ~10 mins to the train station and he charged us 2500 forint (the formal rate). Was a careful driver.
In the train station I purchased 2 1st class tickets to Budapest Keleti.
We entered one of the cafes/restaurants in the station and purchased a couple of tasty sandwiches.
Before boarding went to the toilets - they look straight from WWII...
Boarded the train. The ride was ~3 hours. Comfortable cabin, Saw dozens of deer and huge rabbits in the fields along the way.
Arrived at Keleti and went outside to find a taxi. Luckily we found one within 2-3 mins and it was a 10 mins ride to the hotel.