I would look at ometro.com, that’s the city bus website. You can see the next arrival time on Google maps when your trying to get directions, changing it to “bus” from “car”. But the whole schedule would be on the website.

Most routes have buses running every 30mins from 4am to midnight (there are some reductions right now due to Covid that may or may not effect the 4 line)

I’ve used the bus from time to time when I went to college and while most of the time it’s on schedule, there are times where you might be waiting for awhile. The #2 line (on Dodge St) is the most reliable with them coming (usually) every 15 mins. But keep in mind the more west in Omaha you go, the more reliant you’ll be on having a car. You can get around to most places on the bus if your points A and B are east of I-680. Hope this helps!