Basically all you need to do is have both the old and new microSD cards mounted in your phone -- one being in the phone's integral microSD card slot and the other in an external USB card reader -- and then just copy the contents of the old card to the new card. It's a typical transfer of data from one storage media to another storage media.
But apparently you're having power issues and phone issues that need to be fixed first, then take on the task of that data transfer once you have a working, powered phone again. Keep in mind that when you have an external USB device plugged into your phone, it's not powered by magic. It's using power supplied by the USB bus in the phone, an added drain on the battery. So your phone isn't in it's usual low-power idle state all that time, there are active, power tapping processes in the background (the data transfer between two storage volumes and the plugged in external USB device.)
Once you do get your phone issues fixed, you might need to approach the data transfer issue differently, perhaps in two or three separate sessions, splitting up how much data you pick each session. You can keep an eye on things as they happen instead of just letting things go until the battery drains down too far, and charge the battery back up accordingly in between each session.
Again, with no computer access and you're ruling out using any online storage services, your options are limited so there isn't a really simple and quick solution. It's going to require some extra effort.