How are 10 days calculated when travelling? Would it be 24hours times 10 days? Or is it calculated by arriving relative to salah timings? Ie one Fajr to the next?

My specific example is that I am arriving at a city 31st May at 9am and leaving it 10th June 11am, am I able to fast/pray full?

There are two ways to calculate:
1- If you arrived after Sunset, then the calculation starts from the dawn of next day, till the sun set of the tenth day. Hence, if you will travel before the Sunset, it is less than ten days.
2- If you arrived after dawn, as it is in your example, at 9 am of 31St of May, then the ten days will complete by 9 am of June 10Th.