This happens to me a lot on the train. The entire carriage would be empty except for me (leaving for work before 6am will do that for you), and someone would get on, see all 59 empty seats, and choose the only one next to me.
It's not rude, just rather unusual. They've paid for a ticket and have every right to sit in any empty seat they like. It's not the particular seat I would choose in the same circumstance, but doing something differently from how I would do it doesn't make it rude.
Getting up and moving to another seat, likewise, isn't rude. You have every right to do so, and it doesn't infringe on their personal space, disrupt them, etc. anymore than it would if you were to get up from your seat to alight from the vehicle at your stop.
Of course, if you were to get up and glare silently at your neighbour, that would be rude. And the neighbour would be rude if they sat next to you and then invaded your personal space with their elbows. But sitting in your own seat as you would on any other day, next to another person, is not rude, even if you had other seats to choose from.