I would like to understand, if Facebook gives some free out of the box promotion for new games? I see "New Games" section in Facebook App Center, does every new game gets in there?
The shortest answer is "no", Facebook almost never gives away such placement for free. This is, literally, where they get all their money; Zynga (the big game developer) is the largest source of revenue Facebook has.
While not universally true, a good rule of thumb is that if you see it, its for sale. The largest amount I've ever gotten free was $100 in 'free' advertising, usually for registering for some service or paying for a domain name or hosting plan somewhere, etc.
That platform is available for free, the tools/code/SDK is available for free, the code samples, tutorials, and some forums are available for free; basically the whole development/building side of things is free.
Its the marketing and the advertising that is anything but free. It's very much like a search engine; they'll list you on their own, but if you want anything better you are gonna have to pay.
And there are many, many ways to pay for exposure on Facebook.
However, there are some things that are provided by the Facebook platform that don't cost you dollar bills directly.
To get the free stuff you need to name your app appropriately (like putting "solitaire" in the title), use any tag/description features efficiently with keywords, and apply the right "categories" so your card game doesn't end up listed as a hidden object game.
The other 'free' listing is Friends and Trending, but the only way to get Trending is massive viral explosion (don't count on it, especially without paying for it) and Friends only comes into play as people already start playing your app, helping it slowly spread (but, again, don't count on it).
With permission people can post from your game to their feed and walls, and this is also 'free' in that people do your marketing for you. You can also promote apps from within your own app, but of course that only matters when you have other apps that have a sizable player base.
But as far as big banners, "new on Facebook" promotions, top billing on "suggested for you" pages, etc, that costs money. Big, fat, filthy wads of it. Some features are per view/exposure, like sidebar ads, and some are features they don't even mention in obvious places because they only offer them to people who are already swinging large spends on their sites- kind of like buying front page ads on Youtube.
According to the big players, it costs 50 cents to $3 in advertising to get one new install of a game. That's when things 'work', of course, so it's good to keep this in mind; a free promotion that would get 1000 people playing your game has a market value of $500-3000, and that's a lot to hope to get for free from a for-profit company.
For "guerilla" advertising, where you spend your time instead of money to get the word out on your game, there are lots of resources out there that talk about this; I don't have one I personally endorse off hand, but I think most reputable website/software bloggers do a good job of talking about the basic principles of marketing and how they relate to the game business.
The general idea is to work existing networks of "taste makers" - critics, bloggers, review websites, gaming sites, etc. They don't all require payment to talk to you, and if you are a good communicator sometimes you can talk people into an article, full blog post, etc. YMMV, and sometimes you get little/no traffic out of it.
Bottom-line, "if you build it, they will come" applies to magic baseball fields - not Facebook games, or games of any type. As with all other networks of this type, if you release a decent game its not uncommon to get a few hundred or thousand views/plays in the first month (Kongregate, Armor Games, Facebook, app stores on Android and iTunes, etc), and that can be pretty much it. Anything else takes work + a special game that proves more interesting than about 70-90% of 'released' online games.
But sometimes that amount is just fine! At one point I had the most downloaded/used modification for CivIII on the internet for about a year, and while that 'only' amounted to something like 5-10k downloads across all versions, that seemed pretty darn awesome to me for something I made for fun and just on a forum.
And if that's ok with you, then maybe that's a great start and you can move on to your next project, and maybe if you find a small growing base of players you can use that to help promote your next project. :)