The pronunciation of the Spanish language has changed significantly in the last five centuries, especially what is called the "readjustment of the sibilants", that happened gradually over the XV, XVI, and XVII centuries.
Nowadays, in Spanish, we have only one sibilant, the s. Back in Hernán Cortés' time, there were four: s, x, z, ç, plus ss.
The evolution of the pronunciation can be observed particularly well in the texts of the verbal processes, and in legal documents. Sometimes, you may find serilla instead of çerilla, and things like that.
Vocabulary-wise, if we stick to the basic things, there should be no problems, and mutual intelligibility is assured, with the occassional oddities, like prepositions that are no longer in use. "Cabe" is an archaic preposition meaning "besides", but today nobody uses it. Another quirk is that some verbal tenses are conjugated differently nowadays: " pondría", "vendría", " tendría", "tendrá" have substituted the old forms "pornía", " vernía", "ternía", " terná"...

Long story short, you would have no significant problems understanding eachother, however, you would find eachother's accents a bit odd.

Sources:
Covarrubias, Sebastián de, Thesoro de la Lengua Castellana, Madrid, 1612.
Villalón, Cristóbal de, Gramática de la lengua castellana, Antwerp, 1558.
Gramática de la vulgar lengua española, Leuven, 1559.
Nebrija, Antonio de, Gramática, Salamanca, 1492.
Ruffinatto, Aldo, Las dos caras del Lazarillo: Texto y Mensaje, Madrid, Castalia, 2000.
Las Casas, Cristóbal de, Vocabulario de las dos lenguas, Toscana y Castellana, Seville, 1570.