The figure's FIRST ACTION is a "move action", spends one MP to move on the apostate, thus picking it up. Then the hero/monster may use its remaining 3 MP. And as a SECOND ACTION, it cannot choose a "move action".
by entering the space containing the apostate, the hero/monster cannot move any more, despite having more MP available from the current move action?
I don't have that quest, but heroes and monsters might want to play around that by performing both movement actions before picking up the apostate. This way they would have 8MP stacked(assuming 4 speed), and thus could move 7MP after picking up apostate. Usually special rules like that have some additional rules to prevent exploit like this. For example in 1st quest of Labirynth of the ruin the goblin that picked up the objective would have speed lowered and it's remaining MP would be reduced.