There is no specific word for this.
You could invent one, but then you can't guarantee that the listener/reader will understand exactly what you mean.
What direction is your ticket?
It comes close, but I wouldn't immediately understand what you mean. This seems more like I'm asking which one-way ticket you have bought (A to B, B to A), rather than asking you if you have a one-way or return ticket.
A solution: don't name it.
There no need for you to name this property, because you go on to list the options. Listing the possible options immediately clarifies what you're talking about.
So instead of "type", you could use the context-agnostic word "kind".
a kind of
‘teaching based on a kind of inspired guesswork’
For your example:
He asked me what kind of ticket I wanted to buy: a single or a return.
Note that you can even omit listing the options, if this is part of a larger story, if the continuation inherently explains what is meant:
He asked me what kind of ticket I wanted to buy.
Realizing that I was short on cash, I decided to buy a one-way ticket.
The context makes it clear what was being asked.
Another solution: rephrase.
Since "type" is awkward, and you can't think of a better word to use, why not avoid referring to it?
He asked if I wanted to buy a return ticket.
A one-way ticket is the opposite of a return ticket, so it should be clear from context that you were asked whether you want a one-way or return ticket.
This also omits needing to refer to the "type" in any way, as the context ("return ticket") already makes it clear what is being talked about.