I think you'll either have to spend the night in Rimini or hire a private driver with a van to meet you at the airport. The normal way to get to Urbino from Rimini would be to take a train from Rimini station to Pesaro and a bus from there. However, you could only get as far as Pesaro at that hour. There are trains leaving Rimini station at 10:36 and at 11:30, but there are no buses leaving Pesaro until the next morning. I'm not sure about how you'd get to the train station; maybe you should ask that in the Rimini forum.
You can find train schedules here:
Here are schedules for the buses:
The 25 and 46 routes are the ones that are usually best. The CS and CD also cover the route, but they're circular routes and may not be the most efficient. Many of the buses also stop in Fano, which is the next stop on the rail line after Pesaro, and is also nearer Urbino. If you can catch a bus there, it might be best to take the train all the way to Fano. However, fewer trains stop there than in Pesaro.
Thank you for your message. I spent a lot of time looking through the webpage of Adria. Even if we get to Fano, I am not sure if there would be a transport to Urbino. I also found that it is not possible to rent a car at the airport, all offices are closed at that time. Do you know any private drivers that can provide us with a transfer?
Here is a page with a list of places that do bus and minibus rentals. I used "with a driver" (in Italian) in the search terms, but I'm not sure all of them provide drivers. Rather than calling, unless you speak Italian, I'd send a fax or email, keeping the English short and simple.
I imagine it will be very expensive to have a driver take you all the way to Urbino. It's 65 kilometers.
You might want to ask also in the Rimini forum.Edited: 8:46 am, April 11, 2013
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