Thinking of my own 23-hour layover experience a year ago in August 2009:

(1) recommendations for such a short stay (sightseeing)?

Definitely take #615 city bus. Buy a 24-hour regional ticket for 12 Euros and use it all day. It will be dark by the time you arrive at your hotel. I suggest taking it easy on that first evening. Have a meal in the hotel, in the harbor or in Lasipalatsi.

There are several options the following day: with your 24-hour ticket, take #3 tram or any the trams. Think of it as your hop-on, hop-out transport. Or you can walk in the harbor and have a pre-lunch meal there. Or go to Stockman's and buy your lunch in the food hall down in the basement. Be sure you board the #615 bus back to the airport by noon so you make it to the 2 pm flight to JFK.

(2) will it be too cold to enjoy the city?

You know the saying: there's no such thing as bad weather, only inappropriate clothing. Dress appropriately and warmly and you'll have a heck of a time.

(3) recommendations for a hotel (not too expensive)?

I stayed at Radisson Blu Plaza, a short walk from Central Train Station, where you can catch the #631 bus to the airport. Since my first visit, I've also stayed at Cumulus Hakaniemi and next week at Holiday Inn in Pasilla. Helsinki hotels are not cheap and, as a rule of thumb, weekdays are more expensive than weekends.