In order to create a grammatical sentence, I think that the first line requires at least some sort of word modifying the noun "premier issue". This could be an article, which one site describes as:
Articles are used before nouns or noun equivalents and are a type of adjective. The definite article (the) is used before a noun to indicate that the identity of the noun is known to the reader. The indefinite article (a, an) is used before a noun that is general or when its identity is not known. There are certain situations in which a noun takes no article.
The cases in which a noun takes no article, however, are mostly geographical. For example, the same site says:
Do not use the with: streets, parks, cities, states, counties, most countries, continents, bays, single lakes, single mountains, islands
So one could rewrite the first line of the magazine to be more grammatical, and say:
Welcome to the premier issue of Whatever Magazine!
"The" is not the only option in this case, though, as you pointed out. One could easily replace "the" with "our" and still get a good sentence. This is because a noun usually (see geographical exceptions above) requires some sort of modifier to indicate which "thing" a sentence is referring to. In the original line, there is no such modifier--it could be "our" premier issue, "the dog's" premier issue, or even "the" premier issue.
Edit to address edited question: Even though the title of the magazine is Vegas/Rated, the error still stands. The missing article is before "premier". If the line were instead "Welcome to Vegas/Rated", then it would be an appropriate omission.