Mac Mini

If you have a backup of the old Mini's startup disk, you could use Migration Assistant with that disk to pull in stuff that had made it to the backup.

As far as things in iCloud Photos or iCloud Drive, I think they'd be waiting there for you – once you set up the new Mini to use iCloud under the same AppleID. (If you had a lot of photos in iCloud Photos, it might take a bit of time for the new Mac to synchronize with iCloud.)

if you synced / backuped up the data from your old one, sure you can.