I think it's very unlikely Egyptair will have staffed counters that far before the flight. I wouldn't expect check-in to open until 3-4 hours before departure. That said, if you're able to get Delta to at least tag the bag through to CAI (sometimes they can do this if you show the itinerary) at your origin, then you may be able to re-check the bag at the bag drop-off area on the arrivals level of T4. I'm not sure if they'll require you to have done online checkin beforehand, and I'm not sure if this will work if your ticket requires you to pay for checked bags.

As a backup, I'm pretty sure there's a luggage storage facility (paid) in T4 on the arrivals level. Then you can head into Manhattan via the AirTrain for a nice lunch and some touring/shopping.