Let's see here… well, the moon is climbing the sky, so this poem is most likely taking place at—wait for it—night. Yes, it is true that you can see the moon early in the morning as well, but the moon is rising, which means the action is happening sometime shortly after nightfall. That's about all we get in terms of the details of this poem's setting. We can take a stab at a few others, however. A few poems after this one, Astrophel will be going to bed, so he's probably in his room, looking out the window (or "the casement," as the old Brits used to call it) at the rising moon.
Poor Astrophel isn't feeling so good, and he starts to make some strange claims about the moon: it has a "wan" face, it's climbing the sky with "sad steps," and it is feeling a "lover's case." The speaker is totally projecting his own feelings onto the moon, which makes us think that this poem is also partly taking place in some dreamy, fantasy world where heavenly bodies fall in love, feel sad, get rejected by other planets and stars, and, yeah, pretty much act like people. If this poem were a movie, this scene would probably play some hypnotic music and make us feel as though we were entering some parallel universe.
In addition to these more local settings, this poem is also part of a massive, massive, literary setting: the world of the sonnet. Arguably one of the most distinct literary forms, the sonnet really came into vogue in the Renaissance, when, you guessed it, Sir Philip Sidney was writing. Tons and tons of Renaissance poets (Shakespeare, Spenser, Wyatt, Surrey) wrote sonnets, usually about all the ways in which (usually unrequited) love can cause extreme suffering and pain. While each individual sonneteer has his own particular take on things, it's important to keep in mind that all their poems, collectively, are integral parts of a larger literary ecosystem. Feel sorry for Astrophel, sure, but there are plenty of other palookas like him, pining away though the lines of a sonnet like this one.