This is probably stupid of me to post since it is so obvious, but since it's not in the Mummy's Curse section of Carl's Guide, I thought I'd mention it.
Since the Mummy's Curse is a moodlet, the Moodlet Manager will cure it. I found this out because both my Sims have the Managers. When the guy got cursed, I just simply used the Manager on him (his energy was low) and this cured him instantly. Meanwhile I was still running around thinking I was cursed.
Of course, I haven't been able to get cursed since then. I keep getting the Adventure to be Burnt, Soaked, and Cursed, but only one mummy has appeared since getting it and my Sim beat it in the fight so he didn't get cursed.