Learn Swedish: Felet i artikeln var helt uppenbart. - The error in the article was completely obvious.

In this sentence, uppenbart acts as an adjective describing the noun felet (the error was obvious). Uppenbarligen is a sentence adverb meaning "obviously" or "evidently," which expresses the speaker's attitude toward the whole sentence. You would use uppenbarligen if you wanted to say something like: Felet var uppenbarligen stort (The error was obviously big).

Since uppenbart ends in a 't', is it possible to read it as an adverb in this sentence?

No, it acts as an adjective here. It gets the -t ending because it must agree in gender with the ett-word it describes: ett fel (the error). However, the word right before it, helt (completely), is an adverb modifying the adjective, formed by adding -t to the adjective hel.

If I rewrote the sentence to use uppenbarligen, where would it go?

Sentence adverbs like uppenbarligen typically come immediately after the first verb in a main clause. So you would say: Felet i artikeln var uppenbarligen... (The error in the article was obviously...). Note that you would still need another adjective to finish the sentence, such as uppenbarligen oavsiktligt (obviously unintentional).

Could I say misstaget instead of felet?