Translation:I do not believe that he has already arrived.
'He's' can absolutely mean 'he has', but only if 'has' is an auxiliary verb (i.e. not if it's the main verb).
"He's been waiting a long time" = "He has been waiting a long time"
"He's been getting awfully old" = "He has been getting awfully old"
...but you have to say:
"He has two dogs" (i.e. not "He's two dogs")
"He has a good job" (i.e. not "He's a good job")
Really? After speaking (British) English for 71 years and having just checked various dictionaries online, I am pretty sure that "he's" is a contraction for "he is" and for "he has". Certainly in the case offered above, modern English does not say "He IS already arrived", you say "He HAS already arrived". It sounds Elizabethan (the 1st) to say "He is arrived". No, it doesn't work.
"I do not believe that he has already arrived" is OK, but sounds a bit odd.
"Yet" sounds better, with the negative, as in, "I don't believe that he has arrived yet." (To my ear "arrived yet" sounds slightly more natural than "yet arrived," but both are fine.) Also fine is "I believe that he is yet to arrive," (but probably not as a translation for this exercise).
"Already" is better in the positive, which can be: 1. "I believe that he has arrived." 2. "I believe that he has already arrived." 3. "I believe that he has arrived already."