In Greek mythology, Zephyr (or Zephyrus or Zephros) was the personification of the west wind. The west wind was the bringer of spring and early summer. He also served Cupid (because he fell in love with Hyacinth, etc). Because of this, in European tradition, the west wind has always been considered favorable and mild. On the other hand, in poems like Percy Shelley's Ode to the West Wind, he says
O wild West Wind, thou breath of Autumn's being,
which is kind of strange, because in Greek mythology the situation was the exact opposite - the west wind brought spring, not autumn. In other poems and works, the situation is quite different - see for example Chaucer, who wrote of the "swete breth" of Zephyr, or Shakespeare, who said in Cymbeline
They are as gentle
As zephyrs blowing below the violet,
Not wagging his sweet head.
Your poem seems to agree with the general "west wind is good" interpretation:
Western wind when wilt thou blow
the small rain down can rain
Christ if my love were in my arms
and I in my bed again
In the third and fourth lines here, the author is mourning ("if [only] my love were in my arms") and so in the first line the author is sort of pleading for their luck to change - when will the Western wind blow and return fortune so that their love comes back.
Interestingly, the fact that your poem mentions "my love" is interesting, because Zephyr was also related to love. As mentioned above, he served Cupid, and this was because Zephyr loved Hyacinth, and so did Apollo, but Hyacinth chose Apollo, so in a fit of jealousy when Zephyr saw the two together playing with a discus, Zephyr caused the wind to blow at the discus, which ended up hitting Hyacinth in the head and killing him (stories vary, but some believe Zephyr intended the discus to hit Apollo in the head, not Hyacinth). Apollo was furious but Cupid took him under his protection because the act was out of love.
The Western Wind here may thus also connect to the third line because of Zephyr's longing for Hyacinth, just as the author of the poem longs for their love.