Is it common for a university to buy a plane ticket directly?

You should definitely ask if they can buy it directly for you. I know my department has done this in the past with students who don't have the cash on hand to wait for a reimbursement.

I've traveled several dozen times for university-related business, including several grad interviews. In every case, the vastly preferred method was for me to pay and be reimbursed. However, in almost every case, there was some way around this if needed, it was just much more complicated. Examples have included

• Getting an advance and submitting an expense report later (unlikely for you because you are not yet an employee/student)
• Getting a university credit card tied to my grant (ask after you are a student - I had this as a grad student and it made life much easier)
• Using the department travel card (or having the secretary do it for me if I wasn't trusted)
• Using the university travel agency to book directly

The last two should definitely be options for you.

As graduate program director I insisted that we start buying tickets for all of our interviewees, rather then promising to reimburse them, for exactly this reason. It took a little bit of pushing, but in the end I was able to make it happen. I'm at a big bureaucracy-bound state school, so if we can do it, anyone can.

Your request is entirely reasonable and as a program director I would be very ashamed if my program were unable to meet it.