Well, the cheapest way to get to and from Lima airport (provided you have hand luggage only) is to ignore all the taxi touts, walk out of the airport terminal building veering to the right past the ICAO (!) office, go through the gates and pick up a colectivo (mini bus) for about 1.5 sol. I've done it at all times of the day and night, and often feel safer than using taxis. But then, I do speak Spanish and know Lima quite well.

I don't think the HI shuttle bus goes to other hotels - certainly not the Sheraton which is one of the few major hotels in down town Lima. You can check with them if they do an airport pickup - otherwise you are in the hands of the touts - just negotiate

Henk