I'd suggest you post questions on various forums sites asking for advice while also being sure to mention your site name and why it's better than the competition. Lots of people will see your ad, and you might even get some good advice if people respond.
I'm just teasing. :-) I'll give you the benefit of the doubt.
How do people find your competitor? Is it through Google searches, word of mouth, advertising? Are you planning to get the same users to go to your site instead? Or are you hoping to grow the market for this kind of product? Do you have a hypothesis about who your typical user is, and how they use your site? Are you planning to generate revenue from the site? If so, do you have an idea about what the lifetime value of your user might be? Answering those questions will give you a lot of clarify on what user acquisition strategies to purse.
Buying ads might be effective, but it can also get expensive quickly if it's not translating into enough revenue (or strategically valuable growth). Maybe SEO would be better. Or maybe tweaking the product to give it slightly more viral functionality would be best (I don't know how you would do that for the product you described).
If it's just a side project site, and you're not looking to put much work or money into it, well, good luck.