How much time should I allow for immigration/customs in Lima. Arriving at 4:20 a.m. Then flying to Cusco.?

Usually takes about 30 minutes to an hour. I just got in last night, they-re having problems with luggage, so it took me TWO hours.

I would allow an hour to get through. All the flights from North America at the same time, so even though it's early, the airport will be crowded and you don't want to miss your flight. It may not take that long, but. . .

I would wait until you get there to buy a ticket to Cusco, though, as you should be able to get on a flight if you are arriving that early in the day. You should save money by buying in country. There are flights to Cusco from Lima about hourly.

If you've never been to Peru, though, I would spend a day in Lima, which will permit you to get some rest before heading to the mountains, and to check out one of the major archaelogical museums--National Archaeological, Anthropological and Historical and the Larco Herrerra that is within walking distance, or the Museo de La Nacion

The National Archaeological has exhibits of all of Peru's numerous Pre-Hispanic cultures arranged chronologically . It will help you get oriented to the country and to place the Inca in their context.

I'd say about 1 to 1 1/2 hour. I think that at 4:00am there aren't many flights coming in, but sometimes customs got problems and lines are long...please be patient.
I agree with the first person who answered...you should spend at least a day in Lima to rest and visit the city....there are tons of things to do, depending on what you're interested in....colonial arquitecture, museums, gastronomy, arqueology, or just going around the modern part of the city.

One more thing...as soon as you get to Cusco, try to rest at least half a day before going around the city, if you're not used to the altitude, soroche could really ruin it for you.