My first three mule deer were taken with a .243 it is a very capable rifle if you get good shot placement. I had a neighbor who reloaded his own .243 and I talked him into loading me some 105 grain bullets to hunt with. I took two shots to bring down my first mulie, buck fever almost got the best of me. It only took one shot on the other two. The .243 was my first rifle I ever bought, I've accumulated quite a few since then and just use my .243 on coyotes and pronghorns now.
I would not be afraid to go after a mulie again with the rifle and I would use a bullet in the 80-100 grain range now. My only recommendation is to practice, practice, practice, bullet placement is the key so know how your rifle shoots. The .243 is plenty of rifle to handle deer so don't go buy a bigger rifle, unless you just want one and you have plenty of time to practice before the season.