I think you've muddled your terminals. Lufthansa uses the (nice, modern, clean) T3 with EgyptAir and other *A carriers. T1 is the less efficient, less friendly, terminal used by other carriers.

Your progress between the two depends on whether your bags are interlined. If not, then clearly you'll have to collect them and go through a lot of faff (and probably pay for a visa). If your bags are checked through, or if you are hand baggage only, then life becomes easier. Don't head for immigration, but present yourself at the information/airside transfer desk in T3 with a boarding card for your next flight, or a printout of the itinerary/ticket. Most likely they'll ask you to sit down somewhere while they arrange an airside transfer via the apron by bus. At T1 you face being stored in a buffer zone while they go off with your passport and paperwork to get an OK (and boarding cards) from your airline. When all that's sorted you go through to T1's airside departure area.

The process is somewhat fluid. There are variations at each stage. I've done it a few times, and each time was slightly different. It seems like a muddle, but (like many things in Egypt) it works despite the outward confusion.

Four hours should be a comfortable connection time. Unless there's a significant delay inbound ....

Overnight stays at the cheaper airport hotel ($60 or so) give peace of mind.