There is a high probability that you will be arrested at the airport due to the bench warrant. If you are arrested, they will take you to court and you can explain to the judge at that time. Depending on your explanation and the type of conviction you have, the judge will then either forgive you or add a punishment which may include jail time. I cannot tell you what you will get because I don't know anything about your case.
If you are lucky and you are not arrested at the airport, you should immediately go to court and explain yourself. The same will happen as described above.
Another option is to hire an attorney now to go to court for you and set up a court date, however, it will be extremely difficult to vacate the bench warrant without your physical appearance in court. It may be worth a shot to try.