It has been a loooong time since I was there (almost 20 years), but I really liked Montpellier and thought the city centre was very pleasant and walkable. I am hoping that I'll also get to go next year, but the timing may not work out.
I don't know anything about where the arena is, or the airport, so I can't be helpful with those questions.
But as for things to do, both Nîmes and Avignon are very close. Nîmes has Roman ruins with a very well preserved colosseum and Avignon has the Papal Palace from the time that the popes lived there (13th or 14th century, i think). It was a walled city at the time, and you can walk up on the walls in some places. There's also a famous bridge that only goes halfway across the river. Again, it's been a long time since I've been either place, but both were charming, walkable towns. They are short rides from Montpellier by train or car.
If you rent a car, there's also the Pont du Gard between Nîmes & Avignon. Not sure if it would be accessible by train, but maybe there's a station close enough. I went by car. It's an enormous three-level Roman aqueduct and really impressive to see and hard to imagine it being built without machinery. I think you can hike up to it, but I was just there for a quick stop, so I stayed on the ground and just gawked at it.
I've also been to Carcassone, which is the other way (west) from Montpellier. It has a preserved medieval walled city. I remember it being quite touristy, with a lot of souvenir shops, but I still liked visiting. It's often been used for a filming location. I was there in the 90s for the first time, so I remember people talked about the Robin Hood movie then.