Old Macbook Pro with new iPad

First things first. Getting an iPad Air to go with your old MacBook Pro won't improve the performance or reliability of your MacBook Pro. If you have problems there, you need to figure those out separately.

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As far as compatibility, the current iPad Air's Technical Specifications say that

"Syncing to a Mac or PC requires:

• macOS Catalina 10.15 or later using the Finder
• macOS High Sierra 10.13 through macOS Mojave 10.14.6 using iTunes 12.9 or later
• Windows 10 or later using iTunes 12.12.10 or later (free download from apple.com/itunes/download)"

Apple released several MacBook Pros in 2020. Based on the year and the "2 GHz Quad-Core Intel Core i5", yours appears to be the MacBook Pro (13-inch, 2020, Four Thunderbolt 3 ports.

That machine comes with Catalina and can be upgraded as far as the latest version of macOS – Sonoma. So yes, I would think that you could use that MacBook Pro to manage that iPad Air.

It looks like your MacBook Pro is new enough to support the use of a new iPad Air as a "Sidecar" display.

Use iPad as a second display - Apple Support

System requirements for Continuity features on Apple devices - Apple Support