MM1 definitely has aging, but there's no reason to worry about it. Characters will age by one year each time they level up. They can also be aged by special attacks from certain enemies, notably ghosts. But the most common form of aging is from the more powerful cleric spells, which age the caster when cast.
Of course, by the time you have those powerful cleric spells, you will also have the Rejuvenation spell, which will reduce the age of a character it is cast on. Sometimes it fails and increases age instead, but usually repeated castings will result in a net reduction in age.
Very old characters may suffer reduced stats and can die when resting, so keep an eye on your ages and cast Rejuvenation when needed. I didn't need to worry about age until very late in the game though.
There is also another way to reduce age in the game, which I will not spoil.
Later games in the series made a distinction between "natural" and "unnatural" aging, i.e. your characters had a "true" age, but some enemies could artificially age them. In those games, there were various ways to remove unnatural aging and return your characters to their "true" age. But MM1 and MM2 do not make any distinciton. Age is age, so you can use Rejuvenation to make your characters even younger than they were at the start of the game, if you want.