Which version of Mint are you running?
My Mint system updates Firefox pretty regularly via Update Manager, or (if I feel like going old school, via apt - my understanding is that Update Manager just runs apt behind a nice graphical interface).
Note that Update Manager and Software Manager are different things; The former should be run regularly to keep up to date with security and other updates; The latter is really only needed to install software for the first time, (or to uninstall it).
https://forums.linuxmint.com/viewtopic.php?t=243956