Rise is a loaded term in lots of early modern poetry, but its normally raunchy/sexual pun isn't being used in this case. Here, the word "rise" is literally calling Shakespeare back from the dead (in the form of his printed plays) and is also a reference to his ability to rise above the reputations and legacies of his peers and leave them in the dust.
Thou art a Monument without a tomb, And art alive still, while thy Book doth live, And we have wits to read, and praise to give (22-24)
Jonson's idea that an author's printed work is a living monument to that author is interesting indeed. It also might help explain why Jonson was so eager to see his works into print before the end of his lifetime. Maybe he wanted a monument all for himself?
[…] Look how the father's face Lives in his issue, even so, the race Of Shakespeare's mind and manners brightly shines In his well toned, and true-filed lines (65-68)
Nowadays, teachers are always cautioning you to not assume the author is the one speaking in poems or plays, but Jonson says here that Shakespeare's "mind and manners" live on in his lines. Do you think that means his personality and opinions are in there somewhere, too?
But stay, I see thee in the Hemisphere Advanced, and made a Constellation there! (75-76)
We're not sure about you, but Shmoop can think of a lot of afterlives that would be more appealing than being a star. Elysian fields anyone?