www.venere.com you can 'shop' by location & price. There's booking.com, romeby.com and numerous other sites. Try the search bar above the posts.
I have just returned from Rome where I stayed at the Palazzo Cardinal Cesi hotel. It was really superb, as near to St Peter's as you can get. The bus stop for the open top 110 bus is right opposite the hotel. I have submitted a review but TA haven't posted it yet, maybe within the next day or so. Here is their website, but you may be able to get rooms cheaper from other websites, i.e. Travel Republic which I used.
The taxi from Ciampino is 30 euros, flat rate fare.
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