What causes the speaker's spirits to rise?
I have difficulty seeing where in the sonnet you think the speaker's spirits rise (as they undoubtedly do in Sonnet 29). The mood of the piece seems to be pretty much the same from beginning to end. The speaker is maintaining that he is giving someone longevity through his poem which will outlast monuments dedicated to the most powerful individuals. The final couplet means specifically that the poem will keep the loved one's memory alive until Judgment Day, at which time all the spirits of the dead, including his own, will "rise" from their graves to be judged as to whether they deserve to be sent to heaven or hell.
The sonnet contains a marvelous metaphor in the third and fourth lines:
But you shall shine more bright in these contents
Than unswept stone, besmear'd with sluttish time.
The implication is that a neglected headstone in a cemetery looks like a marble step in front of a home which is smeared because a sluttish housemaid, who should have swept it and then scrubbed it, only mopped it off with water and left it smeared with mud and leaves. It is a metaphor within a metaphor. Time is like that slatternly maid. Most of us have seen old headstones of forgotten deceased in cemeteries and remember their muddy, greenish stains.