I have to assume that your "hour meter" is part of your tachometer instrument and not an external hour meter. Given that assumption - hour meters are simply "turned on" by applying power to them. They have internal "clocks" that then move the digits.
They are not dependent upon any function of a tachometer or an alternator
other than possibly as a source of power or a switch to turn them "on."
All of which leads me to the simple conclusion that the instrument has a partial failure - the tachometer portion works, but, the hour meter portion is not being "turned on."
You can always install a separate hour meter somewhere. Normally they get their power from an engine oil
switch sensor such as your "no oil
pressure" warning light and a relay. However any engine
mounted sensor switch will suffice.