You won't get access waterside but otherwise it's one of the best 'open' ports in Scotland! Providing the weather is good (it sometimes is!)then you can walk from the end of the beach prom (Footdee) all around the outside of the harbour and around to the other side up to Battery Point/Golf Course. Depends how often you stop but I have done it in around an hour to 90 minutes. If walking (or driving) I'd suggest doing it on a Sunday when quayside activity in reduced so you can make frequent stops and peep into the various gates/access points. Just don't be tempted to venture past the gates.
The best place for 'underway' shots is up at the battery but is very exposed in bad weather so you really need a car for that as there is no shelter but it's a popular place when the weather is fine.
Another location is the car park opposite the maritime museum - if you pick the top floors you can get some interesting views of ships virtually in the city centre.
There are also harbour trips - https://www.clydecruises.com/clydecruises/aberdeen-harbour-cruise/
Hope that helps. Drop me a message if you want anymore info.