I started playing FO1 a few days ago and it's lots of fun. But I'm afraid that I won't be able to continue playing (just like in FO3 without BS) after finnishing the main quest (which I think is done by killing the super mutant boss).

I want to do as many quests as possible withouth the timelimit given at the beginning of the game. How do I do that?

If you have the latest patch installed, you'll get all the time in the world once you've finished recovering the water chip. Please not that you still may miss out on a lot of the quests simply because you didn't speak to someone at the right time or insulted someone, or did a quest for one person rather than another.

Also, if you kill the master but don't blow up the vats, you can still keep playing. You can't keep playing after the end of the game though.

The latest patch being 1.2?

I can accept not being able to do some of the quests because well... actions cause reactions. But I don't want to be pressured if you know what I mean.

Won't the Vault be overrun at some point (adding a new time limit) after I retrieved the waterchip? (I'm playing with a guide).

Anything on this will be a spoiler. The patch (IMO unfortunately ) removes the 2nd time limit.
The game wont spring the end on you unexpectedly. (You don't have to read these spoilers.)

And between killing the master and destroying the VATS I won't be under any pressure? (I'll stop after killing the master and leave the VATS alone because I would still like to be able to return to the vault at times).

No pressure, except the challenges of the waste are no match to the slayer of the master.