Should an adverb go before or after a verb?

Both correct, I don't think it's supposed to be limited here. Maybe the creative usage "turns rarely up" would also be used in some cases.

Positioning adverbs is a complex affair. There are some rules of thumb, but for many adverbs, it is quite acceptable to place it before or after the verb. In this case, I think either way is acceptable, though I would probably find the former more natural, i.e.

The word rarely turns up outside of those contexts.

but the following is also acceptable, if a bit less natural:

The word turns up rarely outside of those contexts.

Two side notes: a) you want those (the plural of that), since contexts is plural, b) of is required in both cases here, as it belongs with the word outside.

Frequency adverbs such as rarely go before the verb or divide the verb phrase, as in "I have seldom seen such outrageous behavior." Other examples of frequency adverbs are seldom, occasionally, often, sometimes.

In the first version, the word "rarely" seems to be the subject, rather than "The word," so the meaning is ambiguous.

As in:

The word "rarely" turns up outside of those contexts.