how long is a piece of string?
Exponential growth won't happen, there is a finite market for your product or service and eventually even the most successful site will reach a point of market saturation. You need to consider what market share you believe you will be able to get, look at the share your competitors have, how long did it take them to build that market share? What are you offering to entice their customers to work with you instead.
Where are your 1000 visitors a month going to come from? how are they going to find you? If you are launching an agressive marketing campaign, with a good unique selling point, 1000 visitors in 3 months is not impossible.
Every site grows differently, some get no traffic for months, even years then explode, some do really well from the go, and then never grow, some grow slowly over time and last for years, some go boom and bust. It all depends on your market, your strategy and how appealealing your target market find your site (and this has nothing to do with how appealing you think your site is, if it did every site ever built would be a success)
Essentially, your questions can only be answered by thorough market research and investigation into the needs, wants, and behaviour patterns of your target market. There is no substitute for hard work and research.