Did the clerk mention whether you could pay court costs and get the case rescheduled? Many courts will allow you a one time mess up and it's worth checking on. Otherwise, you may want to hire an attorney to see if it's possible to coordinate your coming into court and taking care of the ticket on the same day, which usually takes care of the bench warrant as well.
If arrested on the bench warrant, you will stay in jail until the next available court date. How long that is depends on which court you are in. Some larger state courts have court daily and move quickly to deal with bench warrent cases. Some courts have "bench warrant" days and deal with them on a scheduled basis. Some municipal courts only have court a few times a month, so you may have to wait a few weeks to see a judge in those jurisdictions.