Yes we did spend a lot of time by ourselves... that was sort of our plan though!
For one thing, we are both 25, and there were not all that many couples our age travelling onboard. This wasn't a problem, just an observation we made. One couple we sat with was right around our age... it was funny how we would constantly bump into each other. I guess we had the same ideas on how we wanted to spend our week. We would joke about how there were 2700 people on board, yet we still ended up seeing them multiple times every day!
Anyway, we had originally been excited about attending some of the classes and programs offered onboard, but ended up just relaxing and doing pretty much NOTHING all week.
The tramway to paradise point was great, and easy to find when you get off the boat. You can see it going up the mountain, so we just walked towards where it appeared to end. Its a very short walk. It was 12 dollars apiece for the tram ride there and back. There were shops, etc at the top but we didn't go to them. We did walk the nature trail.
Also, as far as relaxing and spending time just the two of us, I can't say enough about the Rainforest package at the spa. We got it for one day (Thursday), but wished we had it for the whole week. We went in three times, and two of those times we were the only ones in there. I'm not saying it was private, but it was quiet, peaceful, and relaxing.
Also, I forgot to mention... my boyfriend is a Magic Kingdom cast member.
DCL actually sat us with two other couples who were also cast members - I was the only non-cast member at the table! We were treated just as well as regular guests by every crew member we met. The service was just fabulous. This was my first cruise, and I will never forget it!