How can I appeal outside of the U.S. if my I-829 gets denied?

You continue to be a U.S. permanent resident after your I-829 is denied until a judge determines your residency is terminated. So you can re-enter and you can file an appeal, or even possibly challenge the decision in district court.

You do not need to be in the U.S. to appeal a denial.

As a conditional lawful permanent resident who needs to travel in and out of the country, should your I-829 petition be denied, you can still contest the denial even though you are outside of the U.S. However, as you pointed out, your ability to re-enter the U.S. on your conditional permanent resident card may be adversely affected, depending upon the date of its expiration.

If your I-829 is denied while out of the country, you can certainly return to the U.S. to pursue the appeal.

You do not have to be in the USA in order to appeal. But do you have a valid visa to return to the USA?