How is a fuselage 'puncture' repaired?
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.