I dont see any problem charging by hours, especially when dealing with any project with broad scope that can be constantly change. It's an old argument on these forums, value pricing vs fixed project cost vs hourly billing, but IMO there's no correct answer, so go with what you feel comfortable with. If you're just starting out, hourly is the simplest, less risky option. Value pricing is incredibly hard to pull off, especially when you have equally competent competitors willing to just charge hourly.
I doubt anyone will be able to come up with an 'hours' estimate for you TBH, there's not enough info there and we all work differently. Have you spoken to this client? What info do you feel you need in order to provide that estimate? Try to identify that and get on the phone to them and discuss it. IMO, you need to try to get some idea of budget as well - it's hard, and often requires you to pull out a very vague, wide ranging estimate, but it's going to let you know if you're dealing with a dreamer. It wouldn't hurt to find out how many other vendors they are speaking to (i.e. are you competing with 'like-for-likes', what are your odds of winning when bidding against 30 others etc), and other potential 'show stopper's like projected launch date, any specific technology requirements for the site, whether they are able to pay you in the way you wish to be paid (deposit, monthly payments, milestone payments, etc) and so on. If you can, find out what's most important to them - is it low cost, site functionality, ongoing support, etc?