You can absolutely stay on board all day if you want. The times the recreational facilities (such as the pool) are open may change from time to time, so check your cruise compass (this will be in your stateroom each evening for the next day's port of call), and it will list the hours of operation for the pool and restaurants and bars. However, I don't believe the shops will be open while the ship is in port.
I've said this in other posts, and I'll say it here, just relax and have fun! I know when I went on my cruise on the Majesty I had lots of fun and it was great to just kick back and relax by the pool and do other things after working hard at a new job for ten months at the time I took my cruise.
2007 - Majesty of the Seas (Bahamas)
2009 - Enchantment of the Seas (Western Caribbean)
2010 - Freedom of the Seas (Western Caribbean)
2013 - Golden Princess (Alaska, roundtrip from Seattle)