DOwnload pre bought Lightroom to mac mini

LR5 should run as a 64-bit application under Monterey but installing it is the issue; you must boot into an OS that can support the 32-bit installer (the last being Mojave). That's a big problem for M1's because Monterey, Big Sur, Catalina and M1 Macs do not run 32-bit and can't boot into Mojave. 

IF your new Mini isn't an M1, you are OK but if it is, you're SOL. 

What the others said. There's no way of installing Lightroom 5 on an M1 Mac Mini running Monterey.

You'll have to subscribe to one of the Cloud plans for the subscription version of Lightroom Classic.

https://www.adobe.com/creativecloud/photography/compare-plans.html

Be aware that there are now two completely different versions of Lightroom. What you know as the catalog-based Lightroom 5 is now known as Lightroom Classic.

I see that new MAC MINI's are M1 units.

So no way will that version  of Lightroom install.

Agreeing with @JohanElzenga and adding that you really ought to upgrade to the current Lightroom Classic (which is version 11.4) which will be able to make use of your LR 5 catalog, and provides a billion and 27 new features!

That's a very old version. It's unlikely that it will install on a new Mac.