How is such a 'puncture' repaired? Is the cabin pressurization also affected by such a puncture, and approximately what is the cost incurred to repair such a puncture?
The first step for repairing damage to a fuselage skin panel is to cut out the damaged area1. Cracks tend to propagate, so an area much larger than the damage will be removed to ensure that all damaged material is removed. The stiffeners that are behind the fasteners will need to be inspected for damage as well.
A filler plate will be cut to fill the hole in the skin. One or more doubler plates larger than the removed area will then be cut and all plates will be riveted together. The doublers will possibly be on both sides or at least thicker than the surrounding skin. See here for some examples.
It's likely that a larger repair such as this one will need to be inspected regularly to ensure that the repair holds strength.
It's hard to tell where exactly this damage is located, most of the aircraft skin is part of the pressure vessel. However, the airplane is not completely airtight, so a certain amount of leakage is normal. A hole of this size will probably not prevent the fuselage from pressurizing, though larger ones will.
The cost will depend on the extent of the repair, but could be from \$5,000-\$15,000 (300,000-1,000,000 INR) based on this estimate.
1Actually, the first step in repairing damage to an airplane is consulting the structural repair manual (SRM). This is provided by the manufacturer and describes general methods for repairing minor and typical damage to airplane structure. If the damage is beyond the allowances in the SRM, the manufacturer will be consulted. All of this is to ensure that the repair will meet or exceed the strength and life of the original structure, or if this is not feasible, that the limitations of the repair are documented and understood. Major repairs will require extra inspections to ensure that the structure is still in good shape. Aircraft skin will not be welded, since this can cause heat damage to the metal, requires careful skill and care to do properly, and will not hold up well to fatigue.
For commercial airliners and most pressurized craft, fooot's answer is good. For most light planes, a repair of this type usually boils down to replacing the entire piece of formed sheet metal or fiberglass that was breeched.
An old acquaintance of mine owns a Cessna 150 that she bought from a farmer who'd lost his medical; the plane had been sitting in a field since then until some rednecks came by and took a potshot at the tail with a hunting rifle. She got the plane for a song and also lucked out on a hangar nearby, but the plane couldn't be certified until the dorsal strake that had taken the hit was replaced. From what I gathered it was a fairly simple matter of removing the old pop rivets with an angle grinder and nailsetter, lifting off the old strake, fitting the new one and pop-riveting it back into place, then paint to match. That particular piece was probably the best-case scenario to have to completely replace; most of the rest of the tail is in large sheets and the bullet would have punched clean through two of them requiring a more extensive teardown to remove and replace.
The outer fuselage is not the pressure hull. The pressurized cabins are a different shell and are inside the fuselage. This kind of damage would have no effect on cabin pressure. Even if the cabin hull had been rent by a hole of this size, the effect would be minor. The pressure hull is actually open to the air anyway. Pumps continuously add air to the pressurized area while air flows out through various points. The hole would have to be much larger to defeat the power of the pumps.
To make the repair two things need to be done: the larger open hole needs to be patched and the dent and abrasions around the framing member need to be corrected.
To repair the larger tear, the sheet metal worker will probably cut it out in a rectangular pattern with rounded corners. Then a backing plate will be fitted to the rear of the panel and welded or brazed or riveted in place (depending on the materials involved). Finally a patch will be made that fits into the rectangular recess exactly. This patch will be welded and/or rivetted to the backing plate. The patch will then be sanded smooth and painted. You will barely be able to tell it is there when it is done.
To repair the dent, it will first be pulled out by welding a stud onto the dented portion, then the stud will be pulled outwards. This is repeated until the dent has been removed. Then the small abrasion on the dented area and in the vicinity of the framing member will be filled in using a welding machine. The weld will be abrasively smoothed over, then everything repainted.
The cost to do the repair will depend greatly on where and who does it. In the Boston area I would expect it to cost about $3,500.
If you are interested in the details of acceptable methods of repair, the FAA provides a manual describing typical methodologies: Aircraft Inspection, Repair & Alterations: Acceptable Methods, Techniques & Practices (FAA AC 43.13-1B and 43.13-2B) .