Two items to consider are bicycle fit and riding position. Here is a link to how the position of the controls can effect neck and back pain http://bikeandbody.blogspot.com/2012/02/have-neck-and-shoulder-pain-on-bike.html The post discussed positioning of the hoods on a drop bar bike but I am sure you can extrapolate the information to your own circumstances given your flat bar bike.
An additional thing to consider relative to fit is cockpit room... it is not just a matter of handlebar height relative to the seat. Reach can have a huge impact on comfort particularly with the back and neck. Too much reach has you stretched out and too little room (short reach) has you hunched over.. both situations force your head/neck into unnatural positions to see the road when riding... Top tube length is the primary factor effecting reach but considering you already have your bike you can play on the margins by adjusting your seat fore and aft, as mentioned in the comments you can adjust your bar height, or you can get a new stem that can affect both bar height and reach.
With the proper fit, consider riding form. Your elbows should be slightly bent... that is purposely bent... and your back slightly arched (not hunched). this position has the effect of pulling your head and shoulders up so you can see the road without having to hold your head back. It has the added benefit of opening up your breathing pathways.... elbows bent, back arched should be a mantra until it becomes your natural riding position.
Play with your fit.. you may want to consider getting a professional fit especially if you are considering investing in a new stem or bars. The professional fit will inform you on what would be best considering your current setup.