There are few online tools which allows you to analyze competitor ad space. I would suggest 2 tools SEMrush.com and Adgooroo.com.Both tools will provide very specific information about their organic search, paid search, backlinks, display advertising, top keywords, and main competitors.
There are various way you can promote with small budget but again it depends on various factors which depend on what you want to get out of it. If you are looking to increase your brand awareness than a quick solution is to have display ads on Facebook and Adwords with targeted segment.
If you are interested in discussing more about it than feel free to book for a call to have more personalized solution.
Similarweb.com is another useful tool to see what competitors are doing.
I was helping a guy in the used cellphone resale niche for a day, and he was initially convinced a social media campaign was the next marketing step.
A check of the top three competitors on similarweb showed they are getting less then 1/2 of 1% of their referrals from social media. So this evidence quickly changed his thinking about what would be effective in a marketing plan.
There are many resources for finding out where others are advertising. As for the most cost-effective tool, that's dependent upon what you offer and your target audience. If it's for products, consider Google Shopping.
Note that any advertising is an investment. If you have zero budget, consider other ways to build capital first so you can re-invest it. You'll find that you need to spend a decent amount on advertising first just to gauge exactly how you should be presenting content and bidding.
for Search (Adwords), SEMRush or Spyfu is good. I personally prefer Spyfu these days.
On display advertising, we use www.whatrunswhere.com. They offer a free 3 day trial, and then it's like $300/month, so has to be worth your time.
Would love to schedule a call if you're struggling with researching your competition.