Your bag appears to have a shoulder strap - its perfectly possible to ride with the bag on your back and the strap on one shoulder and under the other arm.
However there are downsides:
The bag will encourage sweat in that one area where it presses.
The bag may suddenly shift downwards, altering your balance.
The shoulder with the strap on it can get achy ...
In contrast to the accepted answer:
of course, stand up
pedal calmly, do not pull the handlebars in (because that causes inscreased knee stress)
sway. Move your body up and foreward, then step and stand on the pedal; repeat
The point is to never have to "pull on the bars and push hard". That's tiring and unhealthy.
It's hard to get a better bike than a mountain bike for going up and down hills. Even a cheap one. The geometry is set up so it's controllable both up and down steep hills, and the gearing almost certainly goes low enough that you'll be able to pedal up even the steepest of hills (as long as you're strong enough to push the combined weight if you, your ...
This is a common conundrum for new bike commuters! Yes, there are certainly bikes out there that will be better for you at hill-climbing or commuting in general. However, it's hard to know if it's worth the investment until you know for sure that you want to continue commuting by bicycle... but it's hard to know if you want to continue commuting by bicycle ...