Can I apply for a transit visa in Oman?

I have a flight landing in Muscat, Oman and another taking off in the following 24 hours. I would like to take advantage of my long layover to exit the airport and actually discover Muscat.

I am looking into the visa options I have and I found something appealing, a transit visa, on the Royal Oman Police website :

Visa type : 23A

Visa name: Airport Transit Visa 1D-3D

Description: Short visit for an airline passenger transiting through Oman on the way to a third country.

However, nowhere could I find a way to apply for this visa. And even though it lies in the "unsponsored" category, another page of the same website mentions that the application should be fulfilled through the carrier (I am not sure I understand the following sentence actually):

To be granted by competent authority at air way passages to expatriate based on application submit by aviation carrier company entitle the holder to enter the country only once and resident for a maximum period of (72) hour provided that he should have enough money to cover his expenses and valid ticket to continue his travel.

Anyway, I searched the website of Oman Air and I found nothing about transit visas.

So, how can I apply for a transit visa through Oman?

I am a European Union citizen. I found out that I can apply for an eVisa for about 1 month but I'd rather avoid paying 20 rials (45 euros) for a few hours of visiting the city.