Overnight layover in Helsinki

We have been to Helsinki, and it's not very large geographically. I would suggest you look on Booking.com for accommodations looking for lower priced properties with the highest customer ratings.
Most would take the train into the center city. Wikipedia says: "The westbound commuter line ”I” runs to Helsinki Central Station via Huopalahti in just over 30 minutes, while the eastbound commuter line "P" runs to Helsinki Central Station via Tikkurila in just under 30 minutes. " It's about 10.5 miles into the center city from the airport.
I have no idea what time you need to be at the Helsinki airport. Your airline should be able to answer that question.
We saw the city on a Hop on, Hop off bus. My biggest memory was that a burger, fries and Coke at TGI Friday's was $60 for the two of us.
Scandanavia is deadly expensive to visit on the ground. We went there on a cruise ship out of Copenhagen.

1) Any hotel within walking distance of the train station would probably be fine. I stayed at Glo Hotel Kluuvi back in 2014 and both the hotel and location were great. We were there on a weekend in August and got a really great rate, but I suspect that prices are much higher in May.

2) I don’t know which is best. I took the Finnair bus. It was quick and easy.

3) I’d check in online and plan to be at the airport by 6:20 (2 hours in advance).

4) With a short amount of time, I’d visit the Church in the Rock (very cool architecture) or visit the Market by the harbor. I don’t think you will have time to do both as I think they both close by 4 or 5. Other than that, the RS book had a self guided tram tour.

I am spending a week in Helsinki in early May. We are staying at Hotel Indigo which is on Bulevardi in the Punavuori district. It will cost around USD $150- per night including breakfast. We are going to catch a train from the airport to city and then a tram to our hotel.

Please disregard my suggestion of Hotel Indigo as our trip has been cancelled due to Coronavirus but they are insisting we pay a cancellation fee of USD $1980-. Our stay was for 26 April to 3 May and their publicised cancellation fee waiver is only to 30 April so they won’t waive the fee because our check out date is after 30 April.

PLEASE DISREGARD CRITICISM OF HOTEL - SEE LATER POSTS. It is Booking.com that is imposing the fee, not the hotel. When I booked it was on the basis that the hotel policies apply, yesterday I was advised that they don't, that Booking.com can override them.

Please see two other threads that AussieNomad began 3 hours after this comment that indicate that Booking.com is not honoring the hotel's cancellation policy and reinforces the importance of booking direct with establishments instead of through third parties.

Regarding the posts above about Hotel Indigo:

It's unclear at this time if the poster being denied a refund chose the non-refundable rate when reserving through booking.com. There is a reason there is often 2 published rates, and customers who choose the lowest, non-changeable, non-refundable one are held to that choice.

Also, the math doesn't work. If, as previously stated, the poster was going to be charged $150 USD per night for 7 nights (26 April to 3 May ) the total tariff would be closer to $1050 USD and not $1980 unless that tariff was pp per night instead of per room? Just curious.

Please disregard my suggestion of Hotel Indigo as our trip has been
cancelled due to Coronavirus but they are insisting we pay a
cancellation fee of USD $1980-. Our stay was for 26 April to 3 May and
their publicised cancellation fee waiver is only to 30 April so they
won’t waive the fee because our check out date is after 30 April.

If you have booked through booking.com you really can't blame the hotel in this case.

FinnAir bus is handy with an underfloor compartment if you have a lot of luggage. The FinnAir bus terminates at the central train station. Train station is centrally located. If you are carry-on only train may be cheaper (but I haven't checked prices).

Hi, an update. It seems that it is Booking.com that is denying the refund, not Hotel Indigo. Please disregard my criticism of the hotel. The hotel cancellation policy would allow a refund, Booking.com won't. When I booked I was told the hotel policies would apply, now hotel says that their policy does not apply as Booking.com overrides hotel policy - this was never advised to me until yesterday.

Also, for person who noted my bad math, my cancellation fee is Euro 1280 not USD 1980 - I inadvertently hit the wrong key when trying to convert AUD to Euro to USD.

I recommend the Hilton at the airport. You can just walk over there after you arrive, drop your luggage, and take public transport into the city.
I would use the time to just walk around and“ get a feel” for Helsinki to see if it is a place you would like to revisit. Just walking the waterfront is memorable. I would highly recommend having food there before you return to the airport Hilton. We didn’t, and my $30 Hilton hamburger wasn’t very good. 😏
With the additional restrictions that will be in place at airports ( probably need to be there at least 2 hours early)I think you will be happy to be close enough to the airport to just walk over.