If you want to visit some tourist-y places, you can go to a Huaca (archaeological site), then eat some typical food at a restaurant, and finally go back to the airport.
A couple days ago I took a friend around for those things. We did Huaca Pucllana in Miraflores, entrance was 12 soles and they have guided tours in English. Tour takes an hour and they open at 9am. After that, I took him to a 'restaurant' in Miraflores Central Park called La Republica, which has typical food and is not that expensive. You may be able to fit those two
