Please tell me the estimated cost of running a dedicated web server of 1Gbps bandwidth?

Well if you are going to rent one from a datacenter then it should be around $150-$200 a month.

Yeah, it really depends on whether you want a low or high end server.

You can get a low end one with an ok processor for $100-$200 a month but high end ones with better processors will cost more (over $300 per month).