Newbie - Help/suggestions please!

Hi all - I'm a newbie to LFGSS but also Fixed Gear riding, so be kind!

I'm a newbie to Fixed Gear riding - so be kind!

With the pandemic and the trains being generally useless, I bought this Fixed Gear bike this morning second-hand via Facebook Marketplace.

At some point, ideally, I'd want to change the handlebars to a bullhorn bar, but other than that there isn't anything else I can think of.

• Where can I get the right bullhorn bar from and are there any specs which I need to find the right one?
  
• The back brake is very worn - can someone suggest me some low-midrange cost to replace them?
  
• Any other suggestions based on looking at the bike that you would typically do?
  
  

Thanks a lot in advance!