Schengen visas come in three varieties:
- A: Airport transit visa
- C: Short-stay visa
- D: Long-stay visa
There was formerly a type B transit visa, but no longer.
The type A visa does not allow you to go through passport control. So a type A visa will not allow you "to leave the airport to visit the city."
(The type A visa is required of nationals of only a few countries, and the list of those who need one varies from one Schengen country to the next. Indians need them, for example, in the Czech Republic, Germany, Spain, and France.)
To leave the international portion of the airport, and therefore to visit the city, you must have a valid type C or type D visa. Since your Schengen visa is expired, you will not be able to leave the airport unless you get a new type C visa.
If you do not have time to get a short-stay visa for your night in Helsinki, you may be interested to read the relevant page at sleepinginairports.net. There does not seem to be a hotel in the international zone of the airport.