The newspaper is going down because they failed to innovate enough to retain value. Taking their model online will not change that
Do you suggest that I not do that at all? If so, what avenue do you recommend I take? Create a Facebook Page for each business? Encourage them to get a website?
To the contrary -- local communities have a growing void in information with the decline or outright death of newspapers -- and filling that can be a great business and is certainly good as a citizen of the community.
You've got two routes in this space: build a local following and become the new information source or help the businesses promote on their own. Since you stated you're new to social / digital, the second seems like a bad idea and a bad offering so the question is can you create content to drum up local eyeballs to build something out of. Don't worry about how to sell ads on day one, worry about how to get people to view them -- there needs to be customers before advertisers come and for customers to come there has to be value in what you are doing.
Your opportunity is in being more than a newspaper. Local culture isn't just a function of what a few paid staff writes, it's the interactions at the bar, the town event for the 4th of July, etc.
Every town is of course different and for some small means community next nothing which can be easy to get the eyes of [but not necessarily so digital], for others it's a community next to something major and thus harder to attract.