Your monster's life span will differ upon what you do to him and the things that happen to him.

Some things that shorten life is:

Fatigue
Do anything that causes fatigue (training, work, exploring).

Battle
It's actually the taking damage part that shortens life, not the fighting. The less damage you take, the longer your guy lives.

Vitamins and Pills
The vitamins and pills are good for strengthening your monster, but they not only shorten the life of the monster that uses it, it also shortens the life of the monster that is created if you combine it with another monster.

Style of Raising
Even though you get more work out of your monster at Hard, they will die sooner.

Some things that lengthen life is:

Upgrading the monster's pen
If you upgrade the pen, not only do they live longer, but they also work better.

Feed your monster meat
Feeding your monster meat after a birthday, a great fight or when they master a new technique will help them live longer.

Style of Raising
Monsters who are spoiled will live longer, but they won't work as good.

Calmstone, Firestone, Wind Whistle and Wind Drum
It's said that if you keep these items in your inventory, your monster will live longer.

Golden Peach and Plant Egg
Feeding your monster these items will extend their lives by years. Although I'm not sure about the Plant Egg, the Golden Peach only made my monster live one year longer. The Peach is the best.