No stopover program unfortunately (they really should, given that their flight timings often leave their Asian passengers with overnight layovers!)
Check out kayak/hotwire/priceline - I usually stay at the 4-star Cumulus or the neighboring Rantasipi near the airport for around 75 EUR - decent prices for Finland. Both have a very decent breakfast buffet included in the price and offer restaurants.
If you don't feel like venturing out into the city (which you should do - Helsinki is definitely worth a visit!) - select one of the half dozen hotels with a stop along the sole (free) shuttle line: http://www.finavia.fi/en/helsinki-ai...sinki-airport/.