Hi Lee,
Like Sedona I think if you stay 2 nights it's probably enough. We stayed 2 nights pre-cruise and went to Tivoli those 2 nights! Actually, my husband had already been in Tivoli in a prior trip but I dragged him back for the 2 nights. It's touristy but very different so I really enjoyed it. We had dinner there as well. We also took a canal boat ride and had lunch in Nyhavn.
The day we were to board Voyager for our Baltic cruise it was pouring so we decided to go to into an art museum located near Tivoli, the NY Carlsberg. What a pleasent surprise it was! I wasn't expecting much but was very impressed with it.
If you have the time I would stay for 2 nights after you get off the ship. Probably the night that you get there and stay on board you'll be busy packing and enjoying your last night onboard and might not spend too much time off the ship ( at least that's what happened to us in our Norway cruise which ended in Copenhagen).So this way you can see everything at your leisure!