Checking a private key is a really compute-intensive task to do by yourself with pen and paper, but you can do this without actually using the private key to send coins.
I would suggest to use a Bitcoin Core client and to create a new empty wallet (shut down the client, move the wallet.dat somewhere else and start it again), then go offline (disconnect your machine from the Internet) and try to import the key as it is described in this question. The client will check the private key, if that's invalid then an error will be reported (I changed a character, it reported "Invalid private key encoding (code -5)").
After this, shut down your client and delete the wallet.dat, move back the original and keep using that.
Of course using this method writes your private key to the HDD/SSD of your machine, therefore it is likely to be recovered by forensic/data recovery softwares. If you'd like to avoid this you will need to use a disposable system, like a live OS that retains no data.
You can for example download Ubuntu and put it on a pendrive then boot from it, by default it will not save anything you do. You can also download the client from the Bitcoin Core download page (choose "Linux (tgz) 64 bit"), save it to the pendrive and then when you boot the Ubuntu you can simply extract it to the Desktop and run from there, all offline.
If you'd want to be more paranoid then before booting the Ubuntu disconnect all other drives from your machine, leave only the pendrive connected. Also, after using the Ubuntu and returning to your regular OS search for a method to securely wipe your pendrive and do that.