Train from Vienna to Budapest

There are many direct trains from Wien Hauptbahnhof (Vienna Main Train Station) to Budapest - Keleti Station. It takes about 2 hours 40 minutes. The fastest and direct are EC and RJ (Eurocity and Railjet). The best option is to buy tickets well ahead (I think two to three months and they are substantially cheaper. Nowadays Flexipass is not a good option unless you travel much more.
For schedule look at Deutsche Bahn website. To buy the tickets go to OeBB (Austrian railways) website.

I wouldn't be surprised if the flexipass costs you more. Check the fares on the national rail services, not on a reseller site. My nephew just bought a train ticket from Vienna to visit us in Budapest Last week, I think he said she paid 30 euro at the station the day prior. Trains Vienna to Budapest run about every hour and the fastest of them take about 2.5 hours.

OEBB, the Austrian State Railway runs trains at least every 2 hours, and there is likely a train every hour. Your problem may be that it is beyond the June 10th schedule adjustment date. All European trains reset their schedules on that date, so until you get within a couple of months of that date, technically, there are no trains after June 10 because they have not been loaded into the schedule computers yet. To get an idea, you can use a date before then to see the schedule and prices. https://tickets.oebb.at/en/ticket is the place to start. Also, www.bahn.com will show schedules, but not prices as it is outside of Germany.

When bought in advance with the Austrian railways (the OEBB website mentioned above), the tickets for Prague to Vienna can be as cheap as EUR 19 and from Vienna to Budapest EUR 14 (plus a few euros for seat reservation) on direct trains.
For the trip from Amsterdam to Brugge you have the choice between the Intercity service and the high-speed Thalys, both with at least one chance in Brussels (Bruxelles-Midi/Brussel-Zuid). The Intercity service doesn't offer advance discount; the Thalys does. Prices depend on travel day and hour. Tickets are best bought with the Belgian railways: www.belgianrail.be/en.

I did pretty much the same route as you a couple of years ago (2014) and I just bought P2P tickets in advance at huge discounts. I also had train rides to Berlin, Krakow, and Munich thrown into the mix and there wasn't any kind of pass that came close to the total $. If you know what days you'll be traveling on and which train times you want, you can get these ahead way cheaper than a pass. The only thing you get with the pass is some flexibility but it'll cost you. As an example my train from Vienna to Budapest purchased in advance was $27. Trains leave Vienna every 2 hrs and take about 2-1/2 hrs.

Do you also intend to take the train from Brugge to Prague? If so, it might be best to travel via Leipzig and Dresden. The train planner of the Deutsche Bahn gives normally a route via Nürnberg, from where it is a bus to Prague.
These tickets can best be bought with the Deutsche Bahn. Because its start and finish are outside Germany, you have to buy separate tickets: one from Brugge to Leipzig or Dresden, and a ticket for the connecting train from Leipzig or Dresden to Prague. Advance bought it can be as cheap as some EUR 60 in total.
It would be a very long day on the train. Why not have a night in Leipzig or Dresden? Both are nice cities. In Leipzig I had once the excellent (and amazingly cheap) Victor's Residenz-Hotel just across the street from the railway station.

As you have likely figured out by now, using RailEurope via the RS site to research schedules and fares is not the way to go. There are trains nearly every hour between Vienna and Budapest.

Noted above, you should use the website for the rail carrier where the journey originates to purchase your tickets and get schedule information. I highly recommend the Man in Seat 61 website for all your train travel questions.

For the Vienna to Budapest journey, the savings fare will be 19 Euro per person if you purchase in advance as soon as possible. Use the www.oebb.at website which has a tab for English.

You should also re-think a pass as it never makes $$ sense these days.

Finally, I would look into flying from Belgium to Prague.

Thanks to all for your comments and suggestions! I will act on them and have a much nicer and less expensive trip!

The only disadvantage of purchasing the 19 euro ticket in advance vs the extra 11 euro to purchase the ticket on the day of travel is that with the advance purchase you are locked into a particular train. If you get bored in Vienna and want to move on a day early you cant do it without wasting your ticket.

you should use the website for the rail carrier where the journey originates
Which is not totally true. International trains are operated by both companies, and you can buy your tickets at each of them.

Yes, but sometimes (as with OeBB) you are required to pick a ticket up from a kiosk. There are no OeBB kiosks in Budapest. I meant it more as a recommendation - thus, the "should."

Doesn't the OEBB offer the possibility of a emailed pdf, as mentioned by the trustworthy Man in Seat 61? www.seat61.com/international-trains/trains-from-Vienna.htm#Vienna-Budapest

Yes, but sometimes no.

Emily's advice is sound. First, there is the issue of picking up the ticket. As an example I have been told that the Vienna to Budapest ticket is a bit cheaper through MAV, the Hungarian rail service, but you would need to fly Vienna to Budapest to pick up the ticket, then fly back to Vienna to use it. But it does work for those who fly into Budapest, go on a tour, the return to Bpest through Vienna.

Then there is the issue of Customer Service should you need it.