How to use get_template part in the plugin?

Yes, get_template_part will always look in the theme. It uses locate_template which is hard-coded to use STYLESHEETPATH and TEMPLATETPATH

I don’t know exactly what you mean about using a “partial template”, but if you mean “can I include part of a PHP file?” then no. It is all or nothing. You need to break your file into multiple files or break the file’s contents into granular functions that you can call as needed.