Before the internet, there were a variety of ways in which you could get tickets for a concert.

Firstly, have a look at these posters advertising the Beatles' 1965 and 1966 shows at Shea Stadium in Queens. Down at the bottom of the poster, there's an information number - which likely would give you details of methods by which and places where you could purchase tickets (certainly as credit cards came into use, you could use them to buy tickets over the phone, but I'm not sure whether that was yet an option in New York City in 1965-66 for this kind of thing). Additionally, the posters both also give details for the Singer Shop Record Dept. in the Rockefeller Center, where tickets could be purchased in person.

Similarly, according to a 2008 oral history of Beatlemania, one JoAnne McCormack discusses queuing at the Cheetah Club in Manhattan from the break of dawn in order to snaffle tickets; presumably, there were also several licensed sellers of tickets across New York's various boroughs.

Before the internet, the likes of Ticketmaster would also have various outlets selling tickets where people would queue up as early as possible in the hope of getting front-row tickets. However, live music as a big business only really kicked into gear in the 1970s (Ticketmaster apparently having been founded in 1976). So in 1965-66, at Shea Stadium (which, The Rutles mockumentary All You Need Is Cash will oh-so-reliably tell you, was named after a Cuban guerilla leader called Che Stadium), instead of a basically-monopolistic ticketing company with outlets, ticket selling in 1965-1966 was bound to be a bit more hodge-podge, based on individual deals with various clubs and shops (e.g., the Cheetah Club and the Singer Shop). I also suspect that you could probably buy tickets at the ticket booths at Shea Stadium where you would ordinarily buy tickets to get into the ball game (certainly you could buy tickets to the Beatles' 1964 show at Carnegie Hall at the venue, according to this poster).

According to a 2013 Jewish Standard article in memorial of Sid Bernstein, the promoter who put on the Shea Stadium concerts, one other option for buying tickets was to send a money order to a post office box belonging to Bernstein. Though judging by the article, this was not the most successful method of getting the tickets, as he had to send about half the money orders back to presumably unhappy customers.