Most players place their Monster Academy outside of the main infrastructure of their base. Most players will ignore them when they attack, and if destroyed, the player will not lose any resources. The only downside to this is the long repair time. Upgrading monsters should be based on both preference and effectiveness. No person should upgrade a monster they will not use often, for it will be a waste of both time and putty. Players often upgrade tank monsters, such as Ichis and D.A.V.E.s, to increase health and attack, and offensive monsters, such as Banditos and Project-Xs, to increase speed and attack. Some of the high-levelled players have also trained their Zafreetis to heal more and have better speed. When upgrading, check the putty cost and stat gains the monster will have. If it doesn't seem as if it is worth it, don't get it.
EDIT: You use the monster academy to train your monsters to a higher level, of which may increase goo costs, health, attack damage, heal (Zafreetis only). Production time may increase or decrease, while housing space does not change except for pokeys, which, after training level 4 & 5, takes up less space. It should be placed in a defended area, as it takes a longer time to repair, and if it is more than 50% damaged, any training will stop.