Do you mean from Spokane to Pullman, or within Pullman to the school?
If you mean Spokane to Pullman, you'll need to take the shuttle I mentioned before. It's about a 90 minute drive, and I don't think any taxi will be willing to take you that far. Here's the shuttle info:
If you mean within Pullman, there are taxis (College Cabs I believe is the company?), but I've heard they're not super reliable. We literally just got Uber and Lyft here this fall, and only a few cars. I've taken them at night, but I don't know if any are driving during the day. It's too small of an area to get much business, so I imagine they typically stick to weekend evenings when they're most likely to get business.
What time is your interview? Classes start here at 8, and the buses are running well before then. If you're staying at the Marriot it's a 10-15 minute walk from the school. They also have a shuttle you can schedule to take you for free I think. The Holiday Inn and Quality Inn are on a main road and should have plenty of bus service from them. Bus is really going to be your best option if you're not renting a car.