How can small businesses/startups with limited budget get press/blog mentions?

Find a niche social media site that pertains to your exact business and participate.
Be helpful, provide relevant and useful information, and your word of mouth advertising will grow from that engagement.
Use the most important social media tools and any content or feature to gain attention or encourage others to link to the website.

Social Media with the great tools will be enough as beginning to you ...

Linkbaiting, which is a variety of practices that have the purpose of generating links to your site.

Its very common in UK for small businesses to get publicity for little or no cost. If your business is something creative or very niche you are very likely to get noticed by the public, they would spread the word through social media and local news press.

Your business needs something that distinguishes itself from others in the same industry. its could be its decoration, service, new technology or cost effeciency.

Most vintage boutique, restaurants, coffee shop, book store and niche products retailers can attract attention from the public with low budget marketing campaigns. Example, arrange an event to promote your products or service, ask customers to try your products or service for promotional price, invite your target audience and they will get the word out. :)