Some good advice here. For little to no money start with online and social media. Check out the books by Gary Vaynerchuk. First what you do is analyze the customer base. Who drinks there? Teens? Young 20 somethings? Older? That there will tell what mediums needed to engage them. For teens to 20 somethings Instagram and Snapchat. Facebook for sure. The advice about being on Google and Yelp is great too.
Then if you have a budget consider traditional media. Perhaps radio and tv? Account Executives in those fields can fill you in on what they offer reaches what demographics you want to market to.
Finally word of mouth? Look at what is around this coffee shop and if there are any direct ways to market them? Is there a large office building or school? Make up some vip cards and menus with nice pictures and put those in areas around those buildings. Most business will have an office manager who might let you leave info in common area like company kitchen. In doing this look at where they're located and perhaps figure where people might be commuting too and from.
If there is a college nearby, perhaps try marketing to local clubs who need a space to hold meetings or small events? Perhaps look into bringing musicians? However some permits maybe needed for that in some places? Some coffee houses near me open their walls to local artists to sell their work and those artists have events as well that bring people in. I would think the coffee house gets a cut from of what is sold.