How much should I charge for half days with a puppy?

That's entirely up to you and the market you're in. For me, I do $30 for a whole day (over 4 hours) and $20 for a half (under 4 hours). For daycare longer than a full day (over 8 hrs or so) I just go up to my full nightly rate. I might charge a few dollars more for puppies than for adult dogs, though I don't take daycare dogs when I won't be home the entire time with them anyway (minus any time I have to spend walking dogs separately, if needed), so the amount of work I put in is pretty much the same regardless of age.

I offer a half day at $20 (1-5 hours) full daycare is $24 (6-9 hours) anything longer than that is the same price as overnight boarding which is $30. I plan on raising my rates when I'm more experienced. I just started on rover at the start of the new year.