Is it possible to go by ship from Djibouti to Egypt?

There doesn't appear to be direct ferry or ship connections as straightfoward as Djibouti to an Egyptian port.

You'd have to cobble together a series of voyages to take you the length, and breadth, as described by The Traveller's Guide to the Red Sea

This vast sea stretches from the Bab Al Mandab strait, separating Djibouti and Yemen in the south from Suez in the north. It is 2,250km long but, at its widest, only 355km across.

By boat, Wikitravel notes:

There are ferry services connecting Djibouti to Yemen. Djibouti City is one of the main ports of eastern Africa so it's well trafficked.

Again, from The Traveller's Guide:

Arab Bridge Maritime (abmaritime.com.jo) operates daily catamarans and ferries between Aqaba and Nuweiba, with fares from US$60 (£34). Ferries also link Sharm and Hurghada, as well as Saudi ports such as Duba and Jeddah from Suez, Safaga and Port Sudan.

Further south, you could haggle for deck space aboard vessels plying between Mokha (Yemen) and Massawa (Eritrea).

Louis Cruises (0800 018 3883; louiscruises.com) runs regular mini-cruises between Suez, Sharm and Aqaba. The Voyages of Discovery 13-day Egypt to India cruise (0845 018 1808; voyagesofdiscovery.co.uk), departing Sharm on 8 December, sails the full length of the Red Sea, with two stops in Oman before reaching Mumbai. From £999, including flights and all meals.