It is possible in one day, but if you only have one day for all of Cartagena, I would suggest you go into the old city instead of the aviary, an hour outside of Cartagena.

The aviary is decent and nice. We had a taxi cab, recommended by our hotel, take us out to the aviary AND WAIT for us while we walked thru the aviary and watched the presentation, then drive us back to our hotel. The drive is about one hour, each way, and we maybe spent two hours maybe at the aviary, so plan on four hours.

Old Cartagena has a lot of history, museums, churches and colorful buildings to look at, as well as lots of shopping, if that's of interest to you. The fort (highly recommend) is outside the old city and was very interesting to go through, but bring some water (though they sell it at the entrance and at the top too), as it will most likely be hot there.

You might be able to see both the aviary and some of old Cartagena or the fort, but you might be rushing yourself to see them all in one day.