Moving SSD from old MBP to "new" refurbished MPB (same model)

I have a mid-2012 MacBook Pro that died (water spill) several years ago. I had previously installed my own SSD. I couldn't afford Apple's repairs, so it sat dead for many years while I used other laptops. Now, I've decided I want to recover all the pictures/files/etc. on the old MBP, so I ordered a refurbished MBP (same model, mid-2012, to save money and to avoid complications).

When my "new" MBP arrives, what is the best way to move the SSD from the dead MBP to the new one? Is it as simple as removing and installing in the new laptop? I'm not sure if I have an "up to date" time machine backup of the old laptop. I'm worried I will get the new laptop and it will force me to format the SSD, effectively ruining my plan and losing all of my files. Has anyone ever come across this scenario?