For comparison, in that period in Spain only about one third of the actors were literate. The rest had to memorise their lines from hearing and repeating them.
So literacy rates are a really tricky thing to nail down, especially that far back. I am an English scholar, so I am sure an historian can come give some more information, but I can give a couple of thoughts about why I think it is reasonable to assume Shakespeare's actors (perhaps in particular) would have a higher literacy rate.
I am not sure where you are getting your 1/3, but I would be curious how they got that number. Spanish theatres at the time functioned differently from their English counterparts, but the largest difference would be the appearance of women on stage. Women in the period were statistically less literate than their male counterparts and I think it is reasonable to assume that would extend to Spanish theatre practitioners.
But for the heart of your question:
England in the Renaissance period was significantly focused on increasing literacy by and large across the country. The shift to Protestantism inherently asked for people to be able to read in order to read the newly translate Bibles. Across the country, literacy was actually fairly high. Greaves cites Cressy as saying there were
Education was being prioritized, especially out of "fear of a general reversion to Catholicism” (Greaves 331). At the grammar and petty schools in the country, schools had literacy as a core expectation for the young boys (and girls when there was space in the classroom). Canterbury Grammar School had as a policy that "No one shall be admitted into the school who cannot read readily, or does not know by heart in the vernacular the Lord’s Prayer, the Angelic Salutation, the Apostle’s Creed and the Ten Commandments” (98 Charlton) (Canterbury Grammar School, 1541).
This brings us to the actors. Shakespeare's plays seem to indicate a shared experience regarding these grammar schools. In addition to the influence of rhetorical texts of the period on Acting, (Curtright has some pretty interesting outlines of this influence in Shakespeare’s Dramatic Persons (Shakespeare and the Stage)) Shakespeare makes numerous references to catechisms throughout his plays. These works (sometimes called absey books) would not only be primers for theological truths, but also would show the alphabet in several different typesets. Shakespeare alludes to these in Othello, King John, and Twelfth Night with characters promising to 'catechize' another. While a small detail, this is likely an indication that catechisms were a commonplace enough reference for not only the actors, but the theatre audience at large.
Finally, we do not have much knowledge of the exact way that Shakespeare's actors would have memorized texts, but we know their primary resource for learning lines was the cue script. This document listed all the lines of an actor with simply a few cue words before each line for the actor to memorize. While I suppose the apprentice actors might have to learn by ear, the rate at which plays would be performed seems to me to make this process prohibitive as primary method of learning lines. Gurr explains that
New plays would be added every two weeks. (Gurr 101). As such, I don't think there would be the strict man power for large amounts of the cast to learn other than by cue scripts.
Additionally, the theatrical space as a whole would be littered with texts, although likely not fiction. Stage managers would make plotting charts listing entrances and exits for characters and post them backstage, a useless act unless the company had a significant rate of literacy (I am trying to dig up the image of the surviving chart, but I cannot find my file for the life of me. Paul Menzer and Tiffany Sterns have some good works regarding early modern theatrical print culture)
So I don't have an exact number for you.^1 I think it would be far higher than we might imagine and certainly higher than the 1/3 of the Spanish.
1 unless you want to me to guess using Cressy's numbers. Actors were technically considered vagrants, but I would bet were probably in line with craftsman rates at the worst. This coming from a perhaps reckless assumption that the relative economic parity and similar apprenticeship systems would put them as roughly equal. But I don't really feel comfortable asserting this.
Charlton, Kenneth. Education in Renaissance England. London: Routledge and K. Paul, 1965. Print.
Gurr, Andrew. The Shakespearian Playing Companies. Clarendon Press, 1996.
Greaves, Richard L. Society and Religion in Elizabethan England. Minneapolis: U of Minnesota, 1981. Print.