In an English fan dub of the OP here, they use "Jesus" as a curse word, but I haven't found anything on Japanese people doing that, so I would assume they could not decipher the meaning and just went with what sounded good.
Jesus is also mentioned in the title so it seems to refer to a person.
Another thing to think about is the "I'm in hell again" line. Is that meant to tie in?
Well, the Japanese word used "地獄" can refer to either the Buddhist hell or the Christian one. (Used in Mark 9:43-44.)
It doesn't seem to refer specifically to the Christian hell.
As the full word was "生き地獄", I think a more accurate translation may be "hell on earth".
It may just be another instance of "use Christian themes to sound cool" thing. Random Christian stuff like crosses and such are added in to Japanese media a lot because it is exotic and strange to them. Using "Jesus" in a song is an example of that I have seen before.
As Arakawa Under the Bridge is very random, it's hard to tell. The perspective of the song appears to change between the stanza and the person who translated it is actually French, and from what I can gather, some bits may not be entirely accurate. However, the French version seems to take some liberties that are not in the Japanese for some reason. I also looked at this translation which varies slightly.
Basically: The song is random like the series, and the use of "Jesus" is just to make it sound cool.
Coincidentally, Hikaru Nakamura, the mangaka of Arakawa Under the Bridge also writes another great series called Saint☆Oniisan, which is a slice-of-life about Jesus and the Buddha on holiday living in an apartment in Tokyo. Probably nothing to do with it, but it's pretty interesting.