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If the port is damaged, then you have to buy an external battery charger and charge the battery using it. This is the external charger you can use to charge the battery by taking it out from the phone --> http://www.amazon.com/Generic-Battery-Charger-Samsung-Galaxy/dp/B008Y8WE5E/ref=sr_1_1?ie=UTF8&qid=1450520952&sr=8-1&keywords=s3+battery+charger
or this --> http://www.ebay.com/itm/External-Battery-Wall-Travel-Charger-Plug-for-Samsung-Galaxy-S3-S-III-i9300-i747-/301723475544?hash=item46401ede58:g:A3QAAOSwNSxVbDJO
This is the only way to charge the battery. If it has some charge, then install Dropbox or Google Drive application from Play Store and backup all your important files to it before it run out of charge.
If you have any questions or would like any additional information, please don't hesitate to ask