rules for standard mandarin vowels:
a: low middle a, like a in "father" but lower
e: prounced like the vowel of "but" unless in combination with another vowel, then pronounced like the e in "bet"
i: prounced like ee in bee unless used with retroflex zh, ch, sh, r or z, c, s, where it's prounced like the i in bit, but shorter and more fronted
o: like the o sound in "boat" but pre-labialized. more like the "au" sound in New England accent (as in "caught")
u: pronounced like oo in boot, slightly less rounded
ü: pronounced like French or German front rounded "i". Say "ee" and round your lips. This vowel appears where marked (lü, nü, etc.), and after j, q, x, y. Because regular u cannot be used in combination with those consonants, the ü is not marked with umlaut. (spelled ju/qu/xu/yu but pronounced jü/qü/xü/yü)
a raises to e (as in bet) in combination with i or y and n: qian is pronounced qien, Yuan is pronounced yüen...etc.
there are more rules...but those are enough to figure out that pu'ercha's vowels are oo, uh, a. an English rhyme with pu'er would be "too fur"
er is not pronounced as "air". in fact, "air/ehr" is not possible in standard Chinese phonology, because strong "eh" only appears in combination with other vowels, and er has only one vowel.
Maybe in some regional accent it's air, but I never heard it pronounced "air" in China, and I travelled to many regions of the country (Northeast, southwest, coastal south, central).