Shouldn't he run as fast as he can, just in case he can get to the other end before the ball is taken?
By this you want to say that a batsman should take a run before he cought out. So they win before he get out.
As per LAW 18 (SCORING RUNS)
9. Runs scored when a batsman is dismissed
When a batsman is dismissed, any runs for penalties awarded to either side shall stand. No other runs shall be credited to the batting side, except as follows.
If a batsman is,
(a) dismissed Handled the ball, the batting side shall also score the runs completed before the offence.
(b) dismissed Obstructing the field, the batting side shall also score the runs completed before the offence.
If, however, the obstruction prevented a catch from being made, no runs other than penalties shall be scored.
(c) dismissed Run out, the batting side shall also score the runs completed before the wicket was put down.
If, however, a striker who has a runner is himself dismissed Run out, no runs other than penalties shall be scored. See Law 2.8 (Transgression of the Laws by a batsman who has a runner).
So even if batmen complete required runs before a stiker get cought out by a filder, they can't win the match.
But yes batsmen should try to complete required runs and should hope a catch to be dropped.
As suggested by @Bogdanovist:
It's hard to judge run outs where stumps are even less than 1 meter away from the crease, then how can they judge catch out and completed runs when the field is away up to 100 meters from the crease.