That's a big jump – 9 versions of the operating system, all the way from Mac OS X 10.9 (Mavericks) to macOS 13 (Ventura). I suspect that much of your software is not going to survive. A lot of early Mac OS X applications had 32-bit code that won't run on Ventura, and even 64-bit applications from that era might have other compatibility problems.
I would suggest going into About This Mac (in the Apple () menu), selecting System Report…, then clicking on the Applications item under the Software heading and waiting for the computer to build the list. Once it has, you can click on the 64-bit (Intel) header (to sort by that field, grouping all of the 32-bit applications together). If you see any 32-bit applications other than built-in Apple ones, make a note of them – since at a minimum, you'll need to upgrade them, replace them, or stop using them.
As for migrating the contents of your old MacBook Pro to your new Air, you can do this with the aid of an external USB hard drive that contains a backup (Time Machine, Carbon Copy Cloner, or SuperDuper!). Your old MacBook Air has USB-A ports and your new MacBook Air will have USB-C ports (also called Thunderbolt ports – because Thunderbolt is one of the things they can carry), so you'll probably need some sort of USB-C to USB-A adapter. (Or maybe a USB-C to microUSB cable for a portable hard drive; there are many ways to do the conversion.)
Migration Assistant will offer you the option to migrate from a Time Machine backup or a startup disk – and that's where the external drive with a current Time Machine backup (or current clone backup) of your old MBP's startup drive comes in.
Transfer to a new Mac with Migration Assistant - Apple Support
You can choose to migrate everything except applications, if you want. If you have a lot of software that will need to be replaced, you might decide that it would make more sense to just migrate your data, and do a clean reinstall of any old, compatible applications (or any new applications) that you need. But that's up to you to decide.