This is highly dependent on your specific city / locality. Sometimes the local jurisdiction will have a preset bail amount for certain crimes, in which case you could immediately post that amount and get out. But not all crimes have these preset amounts, and not all jurisdictions use them. You'd pretty much just have to find out. Failing to pay a ticket is usually a misdemeanor offense, and is more likely to have a preset bail amount, so you will probably be able to get out quickly.
If there is no preset bail amount for the crime, you may have to wait a while. Bail is not set by the jail, police department, or other detention facility. It is set by a judge, which means once you have been processed at the facility, you will be held until you are able to see a judge for your bail hearing. The judge must set the bail amount, at which time you would be able to post bail and get out.
The amount of time you spend waiting varies greatly. In some jurisdictions you will see a judge within 24 hours (if not that day, the following day) and other can take 48 or even 72 hours, but other things can also affect it. A lot of courts don't operate on weekends or holidays, which could extend how long you have to wait for your bail hearing. You will likely spend your time waiting for this hearing behind bars, there's just no avoiding it.
Note that your bail hearing is not your actual court hearing. It is simply a short hearing where the charges are read to you and you submit a plea. If you don't plead guilty or no contest, the judge assigns a bail amount and you are returned to the jail to either wait until your court date or post bail.