I am a Junior Designer in London and have encountered this same problem a number of times now.
I am often asked to style white labelled products (most recently a ticketing platform), which involves creating a stylesheet that overwrites the stock styles of the product with our shiny new ones. Invariably however there is no dev environment or staging server to test my code out and hack around.
My work around / process thus far has been to write my styles in Sublime, then use Chrome extension 'Stylish' to serve my stylesheets after the webpage's. This way my styles overwrite the ones on the page, and can then be sent (minified, of course) to be included in the product.
Though I can style any site this way, it is a slow process as I have to save it in Sublime, copy and paste the code into Stylish, save Stylish and then refresh my webpage. Every time I want to see my changes!
Basically I'm after a more efficient way to edit pages in the browser. Chrome inspector is great but I can't print out the styles afterwards, which has led me to my current work around.
Maybe this is not a common problem? Or maybe my method is totally unorthodox? Either way I wanted to see if any of you encounter similar problems, and how you get around them.