Remaining time on a bounty?
No. Bounties last 1 in game day, or 48 minutes real time. This 48 minute timer decreases as you spend time in free roam in an online session.
The best you can do is estimate or keep track of how many minutes you have spent in free mode since the bounty was placed.
If you are worried about running away all the time, note that time spent in your apartment will count toward this bounty timer.