There are many advertising methods, some are even free (SEO and Social Media channels), so yes, even a small budget can make money if you have time.
If you have a small budget, it’s important to manage your expectations - the volume of traffic you receive and associated revenue is going to correspond with the level of investment you put in.
If you're putting a budget into PPC, results will be faster. However, your marketing campaigns should start small. Assess the initial results and optimise accordingly.
You can then begin to scale-up your marketing activity, knowing that you can acquire traffic at a reasonable cost and that it converts into paying subscribers.
The more optimised your traffic is, the better your return on investment will be (ROI).