Side effects and downtime following PRP injections
The short answer on PRP side effects is that they are minimal. The longer answer depends on which side effects you mean, and on what else is on your calendar.
If you want the procedure in outline first, the Cleveland Clinic page on PRP therapy is a sound starting point.
What the list contains
health.clevelandclinic.org says you may experience minimal side effects like discomfort or pain at the injection sites, headaches, itching and temporary shedding.

Temporary shedding deserves a second look if you are treating thinning hair, because it is a side effect that points the same way as the problem you came in with.
Where the research sits
The wider research on PRP in dermatology is reviewed in the PMC paper Platelet-Rich Plasma in Aesthetic Dermatology: Current Evidence and Future Directions. Read it as background, not as a forecast for your own recovery.
The constraints, in the order they bite
Session length comes first. health.clevelandclinic.org says one treatment can take anywhere from 30 minutes to two hours, so the appointment alone can reshape an afternoon.
Side effects come second. Ask yourself how you would manage discomfort, headaches or itching on a workday in Ypsilanti or wherever you spend it.
Shedding comes third, and it is the one to watch if density is already your worry.
The workable compromise is to book for a day with slack at both ends, then ask the provider what recovery looks like for you. A published list is a starting point for that conversation.
The caveat that stays attached
Read the full article, PRP for Hair Loss: Does It Work?, before you decide.
A clinician who can examine you is the right person to say how any of this applies.