Of course you should! What could possibly go wrong?
But seriously.... It depends on how hilly it is! I've done fixed gear centuries around Long Island and southern NJ, and there's absolutely no reason to be intimidated by that. I've also done a fixed century or two in north-central NJ, where there's a lot more climbing, and that was a whole lot more difficult. There were hills that I could ride up at the beginning of the ride, that I had to walk up at the end.
My bike has a flip-flop hub. I usually use a 17t fixed cog on one side, with a 16t fixed cog on the other side (which I have not used much). So this week I changed the latter to an 18t freewheel. In fact this morning was my first tryout of the 18t freewheel. I felt a little slow; so I think for tomorrow I'll flip the wheel back to 17t fixed, knowing I'll probably switch back to the freewheel later in the day.
It's fairly flat. The course has < 2000 feet of climbing, and much of that is gradual undulations in the roads. There's one daunting climb late in the ride that I might have to walk if my legs are too tired. I have a flip-flop hub too, both sides have a 17t cog, one side fixed, the other freewheel. I've never used the freewheel side. 46t on the front (~71gi). I'll probably ride.