.... when the printer is attached to the computer it just won't print anything. ...
Likely because the DOS OS did not contain a driver recognizing/recognized by the printer or the printer is not working. Hopefully a DOS driver exists for the printer and you can find it. Otherwise, you will have to see what printers the new printer is capable of emulating. Sometimes that allowed other print drivers to be used in a fallback situation. Man it has been a long time since I thought about the 'good old' DOS days.
How you will test the new printer for functionality (unless it has some form of self diagnostic test) is beyond me unless you have a DOS environment (emulation? Even emulations can be problematic with I/O devices.).
Depends upon the DOS version too. And then there is the fair number of DOS projects floating around.
Good luck and look forward to hearing the tale of this adventure into the Jurassic.