Will adding duck/duct tape in my tires stop punctures?
Answer: No, not really.

Duct tape, "duck" tape, or any adhesive tape by itself is not enough to stop a puncture.  At best the extra layers will slow down ingress of a sharp foreign object in your tyre carcass, but no different to a thicker rubber tread.

If your tyre is getting worn in the middle, there's often an uptick in the frequency of punctures which indicates.  Adding layers of cloth or paper on the inside is nowhere near the same as the thicker tread of a new unworn tyre.



Other options:


If the tyre is worn, replace it.  
Confirm your air pressures are suitable.  Too low-pressure means your tyre can't stop the rim from banging into rocks or the road, so you get one or two "snake bite" punctures.  If this is the case, increase your pressure a little, and work on your technique to stop impacting things as hard.
Tyre liners - these are hard plastic strips that lay inside the tyre and around your inner tube.  They are harder and work to deflect inbound sharps.  Downside is an increase in rolling resistance and weight.
Sealant - there are liquids you can run inside your tube/tyre to seal leaks.  They're not guaranteed to work but they do help smaller holes.
Tubeless - if you need low pressure for off-road riding and tubes aren't low enough, then converting or replacing your tyres with full tubeless allows lower pressures.  I have no personal experience of this.
Technique - practice how you ride, and work to avoid the things that cause punctures.  If you flat after landing hard, then stop landing hard.  If you have broken glass in your tyre, then avoid riding through broken glass.




Looking at the spec of your bike, its a 26" with rim brakes and 3x7 gearing with "revo" shifters and sub-Tourney parts.   Its not a high-end bike, so all stock parts will be low-end to meet that price point.  Not a bad thing, but a stock tyre is generally worse than what you can buy new.

The date is hard to pin down but it appears to be a ~2006 vintage bike, so if the tyres are original then they may be cracked or perished.  The tubes can also suffer from embrittlement or perishing.  Same for brake pads.

So the first thing I'd suggest is replacing the tubes, tyres, rim tape, and you may consider new brake pads at this time too given the wheels will be off.