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First Things First!
You must place the "Buy Now" button somewhere on the home page itself that catches the eye of the visitor in a single go. Since you have a "Single Product Store", it will be feasible for you to achieve this. You can place it at multiple pages which means on your Home Page as well as on the Product Page. But, make sure that it doesn't become nagging for your audience.
You can also get your social media pages on platforms like twitter, facebook, LinkedIn, Pinterest and many more in order to increase your online presence.
If you are looking for a one-time solution to achieve both of the aforementioned measures, simply consider choosing a theme which solves the purposes. As your theme is having some minor issues too and does not seem to be user-friendly.
If you need more help, you can Click Here--> https://apps.cedcommerce.com/shopify-store-development/
Hope this helps!