The introduction you made was great and may prove to be valuable in the future. But you are getting ahead of yourself. Let me explain.
It would help to learn more about how that particular school handles graduate student assistanceships first. Call the Department graduate student office and ask about that. How and when do students normally become GRAs? In some schools, it is possible for first-year students to obtain a GRA with a professor. In others, students are expected to work as TAs for at least a year, sometimes more, or even pass a qualifying exam.
Why? You need to learn the time-frame when the professor can make a decision to support you. The professor does not want to waste too much time talking to anyone who is not even admitted to the school yet. If it is a school where students are expected to TA for a year, the professor might not want to waste time talking to anyone who might be "weeded out" later.
I infer that you will be applying to graduate schools this year, for admission next year. The school you mentioned should be high on the list. You will want to find schools with one or more professors whose research interests you, and, more importantly with happy graduate students like the one who introduced you. Financial constraints play a role when professors can take on GRAs, as well as the graduation of current GRAs the professor is supporting.
In short, making the introduction you already did is perfect for now. Seek to make more such introductions with more professors at the same school, and at additional schools. This way, when you attend, you will have multiple options. Once you get to the right place (for example, finishing up your first year as a TA, with good grades in your core classes), it will be easy to talk to the profs you've already scoped out about GRA positions.
If you are accepted to a school where everyone TAs the first year, you can approach the professor about volunteering. This is a great way to get started at the lab. Make yourself valuable and you'll most assuredly get a GRA there when one becomes available.