Is there transportation into the city for a short visit?
http://www.inzagreb.com/inzagreb-tours/zagreb-essential.html. This is a small company that rents apartments and does tours in and around Zagreb. Some on this site don't love Zagreb but I think it is a very cool little city to take a look at. It's not a tourist place, really, but a nice city, great square, great markets, great cafe culture. Since you have so little time, email InZagreb and see what they can do with about 4-5 hours.
I rented an apartment from them and used them for an airport transfer. Very nice people. Good prices.
Yes, there is an airport shuttle.
Inhope u are planning ahead on what u would like to see.
U can just walk around to main plaza , visit farmers market or a museum gem like the museum of naive art. It is a gem imo.
Of course u can taxi back to airport. It is about 10 miles. We pd equiv to about 30 usd airport to our downtown zagreb hotel but always verify with cabbie first. Some are not honest.
One possibility is to sign up with a city tour and arrange your pick-up and drop-off location as the airport. When in Zagreb, I used http://www.zagrebtours.com/en and loved the experience, although I was touring outside of Zagreb, not the city proper. By being a private tour, you'll be able to maximize your very limited time and won't have to worry about getting cabs, figuring out the busses, figuring out the lay of the land in Zagreb itself, etc. The guys at Zagreb Tours were very easy to work with and it was easy to customize the tour to my likings. They should be able to create a custom experience for you.