Both the Manhattan and Melodia are within 15 minutes to the airport but in not very nice areas (but are probably okay during the day time).

When I was in Lima I stayed in San Isidro and took a day tour from my hotel (Novotel). It was a museum tour and I felt it was good, particularly the Larco Museum which has many Moche statues.

For shopping, I highly recommend the Larcomar shopping center. It is set on a cliff above the Pacific and has outstanding views, good shopping and decent restaurants. It is in Miraflores which is an hour from the airport.