Learn Russian: Портной попросил меня примерить костюм ещё раз. - The tailor asked me to try on the suit one more time.

No, this is a great exception to watch out for. While по- can mean 'a little' when attached to a comparative adjective or adverb (like побольше), here it simply forms the standard perfective aspect of the verb просить (to ask). It doesn't mean he 'asked a little bit.'

I usually see requests translated with чтобы (so that). Why does this sentence use the infinitive примерить?

Both structures are correct for verbs of asking, advising, or ordering. Because the person being asked is the direct object (меня), Russian can seamlessly attach the infinitive. Он попросил меня примерить is often preferred over Он попросил, чтобы я примерил because it is more concise.

The word for tailor (портной) looks like an adjective. Does it decline like one?

Yes. Портной is an adjectival noun. It functions entirely as a noun in the sentence, but it takes standard adjective endings for all cases. For example, the genitive is портного and the dative is портному.

I know мерить means 'to measure.' How does the prefix при- change the meaning in примерить?

The prefix при- often adds a sense of doing something for a specific, limited purpose or bringing something into contact with yourself. Here, it changes general measuring into the specific action of 'trying on' clothes to see if they fit your body.