Yep, our one day a week nanny share totally works (probably worth mentioning we were friends before). I brought a booster seat high chair and a nice double stroller I bought off of Craigslist, and he sleeps in...uh, actually I don't know! I believe they had an extra crib.

Nap time: if the host family has a pack and play, they will probably let your child use it for the one day per week. Nanny can set it up in the morning on the day your child comes and take it down at the end of the day.
Where your baby will sleep depends on the home. It might be an extra bedroom, it might be the master bedroom, it might be an office or dining room. Wherever there is room.

You didn't say how old your child is, but for most a travel booster type chair will be more than adequate. You can get one for $25.

It will really depend on what you work out with the other family in the share. In our share, when we hosted, the other child slept in our pack n play and the other family just brought their own sheet. We had two bedrooms and so did they so the visiting child would nap in a pack in play in the parents' room and the nanny would fold up the pack n play after each nap. For a high chair, we had been given a travel one so we just used that when we brought our DD to the other family's home and their DD used it when she came to our house.

For only one day a week, they generally don't work.

I need child care one day per week 8:00-4:00pm and someone suggested looking at nanny shares. How do they work in terms of nap time, would I have to bring a pack n play with me each week to someone elses home and what room in the home would my child typically sleep. Do I have to bring a high chair as well? Just trying to think about how this typically works for a part time nanny share.