When is it proper to abbreviate first to 1st?

Barring cases of extreme abbreviations (where one might use such abbreviations as "t ppl complaind abt t difficulty n reading &c", such as some live internet chat room, or mediaeval manuscripts) then 1st must only be used when first is an actual ordinal; that is it could be replaced by "in position number one" and make the same sense, albeit clumsily:

She was the 1st guest to arrive.

*She 1st greeted her host, then took off her coat.

The first is technically okay, because the first here is indeed an ordinal, positioning her within a series, the second is not because it is an adverb only.

While technically okay, we would generally still not use the abbreviation in the first case, because it's not a context where we would use any abbreviations. Rather, the abbreviations are best restricted to cases where we are going to talk of several ordinals, (i.e. where we would also talk of the 2nd, 3rd, 4th, etc.) or where an ordinal is strongly expected (e.g. in dates). If in doubt, lean toward not using it.