How to do big modifications to live site?

Question

How do you make large site wide modifications to your live site without bringing it down?

I don't have a team of web designers and will be doing this solo. But still need to keep the site up and running.

I'm implementing some changes to the current responsive web design. Some changes will cascade across the site and I'm not entirely sure where all of them might occur. Whole sections of the site might be deformed but I won't immediately know. I don't want to get into a situation where there are too many broken areas and I can't keep up.

I don't have git or a cloud server. I use FTP to move anything or edit the theme through the WP admin area.

How do other solos handle this type of situation?