If the appeal court were to accept the case, he probably gets a stay for the length of the appeal. Of course he may have his appeal rejected and the clock would start ticking again and I believe he then has 29 days.
The appeals court rarely refuses to "accept" an appeal at this stage. I have never seen one which the appeals court rejected an appeal at this stage. The question is if they will stay the order pending appeal. Contempt is a very serious offense, IMHO the only reason the order should be stayed is if E* has demonstrated there are compelling reasons pointing out the contempt ruling is wrong.