How did The Right Stuff get a PG rating with so many uses of the f-word?

Because the PG-13 rating wasn't created until the following year in 1984. And the leniency and context rules for cursing were much more relaxed. The current MPAA rule is a single non-sexual Fuck for PG-13. It is more of a guideline that can be fought, and is done with some frequency.

As mentioned by @cde, the short answer is that it received a PG rating because PG-13 did not exist at the time. Reading this answer led me to some more details that I feel are important.

First: The Right Stuff initially received an R rating.

From wikipedia - "The Right Stuff (1983)"

The film was originally rated "R" (Restricted, which means no one under 17 admitted) by the Motion Picture Association of America because of some strong language (the word "fuck" is used 5 times, which meant a near-impossible chance of it receiving anything short of an "R" rating) a scene of implied masturbation and other hard content; but it was given a "PG" rating on appeal (the PG-13 rating did not exist then; it was created the year after this film was released).

Second: Only two movies have ever been rated PG that used the f-word

From wikipedia - "List of films that most frequently use the word "fuck"" -

The Right Stuff (1983) contains five uses of "fuck", and All the President's Men (1976) uses the word seven times. Both portray historical events and were granted PG ratings on release; the PG-13 rating was introduced in 1984. The Right Stuff continues to have a PG rating. The current version of All the President's Men contains 11 uses of "fuck" and is rated R.

So the only other PG movie that has ever received that rating while still using the f-word was "All the President's Men" which received an R rating in its current version because the content was changed (now contains 11 uses of the f-word).

So the answer appears to be that The Right Stuff received a PG rating because of an appeal which was apparently granted because it portrays historical events, and the PG-13 rating did not exist at the time. Most likely, if PG-13 rating had existed at the time, it would have received that instead of PG after the appeal, but that is only my personal opinion.

An obvious followup question as to why the rating has not changed is because the content has not changed and a rating appears to only be changed either due to a change in content or due to an appeal prior to its release.