It depends on the app. Some need 4.0 minimum, some need 4.1 or 4.3 or 5.x. Some will work only on certain hardware.
It probably wouldn't hurt to update the phone, since it was made to be updatable to ICS, but don't count on too many currently available apps working with it - it's over 3 years old which, in Android terms, is about 4 generations old (like a 100 year old person).