Maykapar spent his childhood in the city of Taganrog and in 1885 he graduated from the Boys Gymnasium where he studied with Anton Chekhov.
He also took private music lessons from Gaetano Molla, director of the Italian Opera in Taganrog.
A pupil of Beniamino Cesi and Nicolai Soloviev, Maykapar received the St. Petersburg Conservatory's diploma in 1893 and further improved his piano skills under the direction of Theodor Leschetizky until 1898.
During this time he gave numerous concerts in Berlin, Leipzig, Saint Petersburg, Moscow, Taganrog and other cities.
Maykapar was an active participant and the Secretary of Sergei Taneyev's Scientific-Musical Circle in Moscow.