Two related places spring to mind, which are only a short distance away from the city centre. You can cycle to them, and do them both on the same trip, so that might help!
Firstly, I'd suggest you head to Stanley Park. If you head out to the easterly point, you get great views over the city, the harbour and the mountains. It's not very high here, but nice views!
Next, carry on round (it's largely one way round the park for bikes), and head up onto the Lions Gate Bridge. (You need to come a little way inland to get on). Pick the right hand side, and follow the cycle route onto the bridge. Pause around the first pylon, and you're now rather high up with stunning views!
Finally, head back the way you came, and then round the western side of Stanley Park. You're quite high up initially, so you can get nice views over English Bay and to UBC.
I believe there are also some trails in the mountains which are free to go up. The maps around Lynn Valley certainly suggested it, but I've only done the bits around the canyon bridge there, didn't have time to go much higher...