Short answer. One year. Article 92 of the Revised Penal Code provides that light penalties imposed by final sentence — the category that includes arresto menor — prescribe in one year if never served, the shortest prescriptive period among all the penalty categories the article lists.
What the law says
The penalties imposed by final sentence prescribe as follows: 1. Death and reclusion perpetua, in twenty years; 2. Other afflictive penalties, in fifteen years; 3. Correctional penalties, in ten years; with the exception of the penalty of arresto mayor, which prescribes in five years; 4. Light penalties, in one year.
Revised Penal Code, Article 92 — When Penalties Prescribe. Read the full provision →
Light penalties prescribe in one year
Article 92 lists the penalties imposed by final sentence and how long each takes to prescribe, and light penalties are given the shortest period: light penalties, in one year. Arresto menor falls within this light-penalty category, so once a sentence imposing it has become final, the one-year period set by this article is what applies if the sentence is never carried out.
How this compares to the other categories
The same article sets longer periods for heavier penalties: fifteen years for other afflictive penalties, and ten years for correctional penalties, except that arresto mayor — itself a correctional penalty — prescribes in five years instead of ten. Death and reclusion perpetua prescribe in twenty years. The pattern across all four categories is that the more severe the penalty, the longer the state has to actually enforce it before the sentence prescribes; light penalties sit at the opposite end with the shortest window.
The death penalty reference is superseded
The article's first item still refers to the penalty of death, but Republic Act No. 9346 has since prohibited the imposition of the death penalty in the Philippines. Where the Revised Penal Code's terminology is used, reclusion perpetua now takes its place; where it is not, life imprisonment applies instead. This does not affect the one-year prescriptive period for light penalties that answers your question, but it matters if you are reading the rest of this same article.
What prescribes means for a penalty
Article 92 is about the penalty itself expiring through the passage of time after a sentence becomes final, if the state does not act to enforce it — a different concept from the prescription of the crime itself, which is about the time limit for prosecuting the offense in the first place. For a light penalty like arresto menor, once the one-year period specified here runs out without the sentence being served, this article addresses the penalty's own prescription rather than whether the case could have been filed at all.