Yes it needs a new battery, any phone shop will fit a new one for you.
My wife has the same phone, I took it to the repair shop and they fitted a new battery while I waited , only took 2 minutes cost £29.
The repair man said that he thinks the iphone has a worse battery than other phones as he gets more of them with a duff battery than any other make. You need a special tool to open an iphone to get the battery out, I think it is a con by Apple to make more money for phone repair shops.