Two years ago I trailered my '62 from Lebanon, MO (rather hilly) to Detroit with my 2000 4-door Blazer (4.3 liter V-6, auto transmission) on a U-Haul dual axle car trailer (with brakes) and had no problems, other than the trailer wasn't so great and one of the four trailer tires was practically bald after 700 miles.
Even in the hilly areas in Missouri, the truck rarely down-shifted to get up a hill at speed. In fact, I think it only down-shifted once the entire trip. I went 65-70 most of the way but made sure to leave plenty of room between me and the cars in front of me. Braking distance seemed to be about double, although I never tested the limits. Remember to keep it in third gear as opposed to overdrive.
Also, if you're renting from U-Haul via the website or 800 number, be aware that they may not be willing to rent you the trailer if you tell them you're towing a C1 with an S-10 Blazer. Per their books, contrary to what your door sticker says, the S-10 can't handle it. I can't remember their logic (I think it has something to do with the weight of the towing vehicle versus the weight of the vehicle being towed) but a buddy of mine at Jeep Truck engineering said the Blazer is perfectly capable of towing up to and slightly beyond it's towing rating, despite what U-Haul says. So, when I called back the second time, I told U-Haul I was towing an '86 Honda CRX and they didn't blink an eye. The people at the location where you pick up the trailer will never know (and probably don't care) because you won't have both cars with you when you pick up the trailer. If I recall correctly, you can use the online rental tool to identify the approved combinations.
Good luck.
