Glad to hear you kept on playing. Sounds like you caught up pretty fast.
It sounds to me like you are doing everything you can. As Persia, you get cheap Sci improvements, so make sure you have universities in all your major cities of course. If you can get Theory of Evolution, obviously, that's a big help, but since Egypt got that tech first (I'm assuming) he may have a jump on you there. You might wanna investigate his city and see where how many turns he has left and see if you can maximize production in your city to beat him. While you investigate, check out what govt Egypt is in. If it's Democracy (which it probably is), any war is gonna cause them problems with WW, force them to switch govts, and that will kill their research.
I have no idea what your world looks like. Is Egypt right next to you? Are there civs next to Egypt but not to you? Say, for example, you are on one continent, Egypt another, and each of you has one other civ on their continent. You need to help the other civ get powerful enough to cause Egypt problems, and to keep Egypt from getting more powerful by killing them off. Hold off until you get Steam Power, and see if you have coal. If not, you should be able to trade a tech for it to somebody. If you have to trade for coal, make sure you have tons of workers so that your whole world can be railroaded within 20 turns!
My bet is that Egypt will research Steam Power, Industrialism, Communism, and Espionage. I usually do the first two, then switch to try to get to electronics line while building factories and coal plants. I'm sure other people do it differently

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Just remember that any Civ that doesn't have Saltpeter is still using Pikes to defend. If you give them Nationalism, they will be a much tougher nut to Egypt.
Good luck!