On syntastic repo https://github.com/vim-syntastic/syntastic#installation, the only example they provide for installing syntastic is using pathogen.
How should install it without pathogen or any package manager?
Pathogen is only needed for Vim 7 or earlier. In Vim 8 all "Pathogen's" functionality is already built-in, and you should never use it at all.
To install a plugin in Vim 8 you only have to put it under the following path: ~/.vim/pack/<bundle>/start/<plugin-name> Here <bundle> can be any name of your choice. Then restart Vim and issue :helptags ALL to rebuild help tags, and you're done.
There still remains a question how you'll be updating your plugins, but you can do it manually with git etc.