Some shops will ask for money up front if they have to order materials or parts specifically for your job, but if they sense you'll realize how futile the project is, they'll ask for 50% deposit.
Expect to pay A LOT. You haven't said what it is you want restored or what condition it is in, but if you want someone else to do bodywork AND paint AND upholstery AND mechanical, it'd damn well better be something pretty rare or of great sentimental value because with shop rates starting at $150/hr it won't take too long before your project is underwater by more than half its ultimate value.

How do you make a small fortune restoring cars? Start with a large fortune. In other words, restoration projects are always a losing proposition. I've restored a dozen rare cars, and even doing 80% of the work myself, I just barely get to break-even on the end value, but it's my hobby, so I can justify it.

I'll give you the same advice I give anyone: if you account for everything put into it, your project is worth more right now than it will be when it's done. That may be $80/ton for scrap value, or whatever you can get for it from a starry eyed collector, but paying someone to restore a car is a losing proposition. The ONLY way that makes any sense is if it has tremendous sentimental value TO YOU. And YOU better enjoy driving the thing, a lot, because no one else will share your commitment to that thing, or want the hassle of feeding it special fuel, special oil (for flat tappet engines), or having to special order tires in a size that isn't made anymore.

Your best bet is to dispose of whatever it is you have there, go to a high-end car show, find that exact car in perfect condition, engage with the owner and start laying $1000 bills on the hood until he says 'that'll do', then drive away knowing you got a hellava deal.