What's the best way to get from Vieux Lyon to part-dieu station if it's very late at night?
If it's after midnight, taxi or Uber. You can walk as well, it will take you roughly 45 minutes.
Taxis are available at the stands, where they can realistically get a client. I wouldn't count on just walking to the one of the listed taxi ranks and be sure to find a taxi at that time of the night. Just call it or ask the barman (if you're at a bar) to call it for you. You can book a cab in advance or just call before you're ready to go. The two biggest taxi companies are
Allo Taxi - 04.78.28.23.33
Taxi Radio - 04 72 10 86 86