Broad brush picture, for the FFG/CRUDES/LPD side:
O-1: Duty INCONUS 6/wk. If you've (the command) been bad, it can become 4 or 3/wk. Like when the entire LANTFLT somehow collectively forgot how to not have ND's with the M240: http://hamptonroads.com/2010/09/accidental-gunshots-navy-ships-prompt-review
You'll stand OOD Inport. Man the quarterdeck once during the duty day for ~5hrs.
O-2/3: INCONUS CDO, you've got the ship at the end of the day. Depending on how liberal CO is in designating CDOs, you can see it rotate between up to 17 CDO's (good deal) to just the DH's (just terrible).
Rumor always was big decks have more sections to work with.
Standing watch underway would be the closest equivalent to flights, so yeah, at least once a day if not more. That basically is the primary non-administrative reason you exist. Bridge, CIC, or engineering.
Inport duty is probably the shittiest part of being a SWO ENS.
Agree, and since some on this board have no idea what this means... ENS have duty 1x/week with a shitty watch. JGs have duty 2-3x/month, no watch. Again, this is in port, underway is a different beast.