Okay, I'm looking at a keyphrase with a good number of uniques. My site is made in 2007 with 3800+ backlinks. Site at top position for the keyphrase, made in 2000, 209 backlinks, 1 from DMOZ. If I try for this keyphrase and submit to say a couple hundred directories in a couple days, how long would it take me to achieve first position? The other rankings for the keyphrase are all low backlinks, semi old sites.
My experience has been that, under the right circumstances, I can get in the top 10 very quickly, but once my site gets assigned a pagerank, the site may decrease in its rank during the "Google sandbox effect." My guess is that Google is reasoning like this: A new site could be important, so it should be judged to have at least some value in place of zero pagerank, but once Google knows what pagerank it should have, the pagerank is factored in. I realize this doesn't precisely answer your question, but since we don't know a lot of information about your site (how much content it has, the pagerank of its backlinks, etc.), and since Google's formula is partially secret, it's difficult to predict the outcome with any precision.
If you have an active blog and post an article about your site with your back link and ping it you can possibly get listed in Google within a week. I personally experienced a top position on the first pages of Google for several keywords in only 2 days. This may not be common for others. These days the number of back links isn't as important as the content to Google in my opinion. Good Luck