I'm aware that commercial pilots require special training for landing at Category C airports (e.g. Innsbruck), but I was wondering if, as a private pilot I could land at an airport such as this, whitout training provided I feel comfortable enough to do it.
Categories are for commercial pilot operations, they don't apply to private pilots, so there's no inherent regulatory limitation to flying into a Category C airport. The regulations say commercial pilots must be appropriately trained, and airlines have specific rules internally for this and say who can land at them, i.e. captains only or not.
A private pilot can land at any of these, provided the airspace and airport rules allow it (see @LittleCode's answer for some of those), it's up to the pilot's judgement to train themselves. Keep in mind that for many of these airports the special challenges of landing or departing for a light aircraft are mitigated due to much slower approach, takeoff and landing speeds, especially in visual flight conditions. In instrument conditions these airports can be very challenging to get into or our of, but you can still do it if you have an instrument rating, currency and the guts to do it.
The categories you are referring to fall under a classification system internal to airlines. Related: Why is Heraklion airport a 'captains only' airport?
Any restrictions imposed by the national authority will be included in the aeronautical information publication, or AIP. Taking your Innsbruck (LOWI) example, here's what the Austrian AIP says:
184.108.40.206. Operation in VMC on site or in a flight simulation training device FSTD (full flight simulator-FFS; Flight and navigation procedures trainer II-FNPT II) is required before first use of the approach procedures in weather conditions of less than 3000 FT (AAL) Ceiling and 5 KM Visibility and for the approval of any special approach and/or departure procedure.
NOTE: operation in an FSTD shall include the program in VMC as well as in IMC unless a collision detection system is used.
— Source: LOWI AD 2-1. Austrian AIP. 25 MAR 2021. [emphasis added]
The AIP goes on to list further details. Based on the excerpt above: if weather minimums are below those specified, prior experience in good visibility or in an approved flight simulator is required.
Generally speaking, the short answer is "probably not" (I don't know Innsbruck).
If you look into it (as a private pilot, you should know "how" already), you will most likely find that there will be a number of obstacles in your path at the majority of large airports.
These obstacles can include (but are not limited to) :
- Airspace restrictions (both local area and at the airport itself) so, for example, you might need IR rating and a suitably equipped aircraft.
- Technical limitations for your aircraft (e.g. your aircraft might not be able to meet speed requirements)
- Airport prohibitions, for example Heathrow publish the following:
Flights for recreational, commemorative, charity and record breaking purposes, light twin engined private aircraft and all light single engined aircraft will not be permitted to use the airport. 'Light aircraft' shall be defined as any aircraft that has a maximum gross take-off weight of 12,500 lb (5,670 KG) or less.
And even if the immediate technical obstacles don't apply, you might find commercial obstacles such as (but not limited to) :
- The requirement to nominate and use a handling agent for ground ops
- Very high landing fees
If you have questions regarding the specific Airport it is always better to check in AIP. For Innsbruck, for example, VFR (most of the PPL pilots), Airspace D, and no restrictions mentioned in Airport info, so yes it is allowed.
And you can find Visual Approach Chart in AIP.