Is “attempt” a durative or a punctual verb?

It may be either. I quote from grammaticalfeatures.net:

Punctual and durative - these refer to situations which are not conceived of as lasting in time (punctual), versus situations which are conceived of as lasting for a certain period of time, however short it may be (durative). Inherently punctual situations can be further interpreted as semelfactive (taking place only once) or iterative (repeated).

To attempt a field goal in (US) football is punctual, and indeed semelfactive, and may be, in the course of a game, iterative; to attempt to build a consensus behind a policy is durative.

The punctual/durative contrast should not be confused with another pair of aspectual terms which I believe are in play here (same source):

Telic and atelic - these refer to situations which have an internal structure consisting of a process leading up to the terminal point and the terminal point (telic), versus situations which do not have an inherent endpoint (atelic). In this semantic distinction, it is particularly clear that situations are not described by verbs alone, but rather by the verb with its arguments (subject and objects), and it is in fact difficult to find sentences that are unambiguously telic or atelic. The telic nature of a situation can often be tested as follows (Comrie 1976:44-45): “if a sentence referring to this situation in a form with imperfective meaning (such as the English Progressive) implies the sentence referring to the same situation in a form with perfective meaning (such as the English Perfect), then the situation is atelic; otherwise it is telic. Thus from John is singing one can deduce John has sung, but from John is making a chair one cannot deduce John has made a chair. Thus a telic situation is one that involves a process that leads up to a well-defined terminal point, beyond which the process cannot continue.”

Note that this author ascribes aspect to situations and the utterances which describe them, not to verbs. In my opinion this is the most convenient way to treat not only aspect but tense and mood in English. My opinion, however, is not expert; for a more authoritative treatment I refer you to the link above and to this ELU question and its answers, particularly that by Drew Ward. His CALLE blog has a valuable series of essays on Time & Language which is still in process.

I attempted for nearly five minutes to perceive it as a punctual verb but finally gave up.