Can I use my "old Passport" (used for old MacBook Air) on my new MacBook Air?

You should be able to. Depending on the. age of the old MacBook Air, it may have had USB ports instead of USB-C and you may need an adapter such as USB-C to USB Adapter - Apple

Depending on how old the "old Passport" drive is, you may want to consider getting a new one. Depending on how old it is, it could be living on borrowed time. The backup device should be the most reliable part of your system, otherwise what is the point. You could still use the old one as extra storage space as long as it is backed up.