Prepositions are a common source of mistakes for learners of any language. The word “Internet” is especially problematic for an English learner because it can cause several problems at once. First, when it is used as a noun describing the network we all use, it is used with the definite article:
I wrote “dubious” instead of “wrong” for the second option because some native speakers do use the noun without the article. However, most English speakers consider the first option to be the only correct one, so it is the one you should use.
The word “Internet” isn’t preceded by an article when it is used as an adjective in front of a noun that itself takes no article, e.g.
The term “Internet access” is sometimes shortened to “Internet”, in which case we don’t use an article:
The second sentence doesn’t make sense; you can’t own the Internet, but you can have Internet in the sense of Internet access.
Another problem is the preposition. When something is part of the Internet, we say that it is “on the Internet”, not “in” or “at” the Internet:
As for whether you should capitalize “Internet”: It’s hard to make a mistake here. Both “Internet” and “internet” are commonly used when referring to the network. Traditionally, “Internet” was considered a proper noun and written with a capital letter. Nowadays, the noun is considered to be a generic name, like “electricity” or “water supply”, and is commonly spelled “internet”, so:
Some style guides prefer one spelling to the other, so if you are writing a text for someone else, you may want to check which style guide they follow.