[anvandare]

The summer solstice ("the longest day of the year") is always the same date regardless of your latitude: June 20th/21st (in the northern hemisphere).

And assuming Syene is on the same longitude as Alexandria[0], Syene's noon is Alexandria's noon.

  Alexandria:       31°11′51″N    29°53′33″E
  Syene:            24°05′20″N    32°53′59″E
  Tropic of Cancer: 23°26′10.4″ N

[0] It isn't, which contributed to the measurement error, but close enough is good enough.