How can I back up my iPhone without Wi-Fi? I assume there is something wrong with the battery in my iPhone 4, because it will not stay on unless it's plugged into the charger. 

I have about 1,000 pictures on my phone, and when my phone is plugged in, I can look at them, and still use my phone somewhat normally but my phone will not connect to Wi-Fi. 

Is there any way I can keep my phones pictures if I have to get a new phone, without doing an iCloud backup?
You can back up to iTunes on your Mac or PC desktop or laptop directly over USB.

Connect the device over USB to iTunes, select your device in the toolbar and click Back Up Now.