I haven't been the whole way but I can in part answer your question. It looks on paper to be close and short (just 30 something k but don't be deceived it will take up to several hours. This is because a) you have to cross the river. Ferry crossings can involve long waits. There is a new ferry now so if you go on that it is mean't be alot safer, but may require a long wait. b) roads will take longer to drive on. c) you have to cross the border. The Gambian Senegal border is by all accounts easier than many and I havent crossed at that point and not at all for a long time, but it is a border and always adds uncertainty to your travel plans. In short if you want to go to the game park then I would recommend either going with an organised group (when they do the logistics for you, but will cost more and may be not what you want to do) or staying somewhere North of the river for a a couple of nights (we used the Bradt guide and found it brilliant so perhaps look on there to see recommendations of accomodation near Barra/ Fass, perhaps taking in the Nuimi national park area too while you do). Leybatto is well located for Banjul, Serekunda Bakau and traveling a little inland too. For example Tumani Tenda (not to be confused with Tendaba) the stunning little remote eco village is 2 hours away and is a great pl;ace to stay for a couple of nights too.

There will be a number of 11 seater Toyota mini buses at the airport - just walk up and haggle away. it should be about £60/ Eur 65 depending on the exchange rate - just agree a price beforehand.