Q: Is there a link containing S. Kato's 1978 preprint "On an explicit formula for class-1 Whittaker functions on split reductive groups over p-adic fields" ?
It seems the answer is no.
I have inquired with one of professor Kato's collaborators, professor Moriyama from Osaka University, who could give some information on this manuscript:
The paper you mentioned is the master thesis of S-I. Kato under the supervision of Prof. N. Iwahori, which is hand-written and written in Japanese. I read it in the library of Graduate School of Mathematical Sciences of University of Tokyo more than 20 years ago.
The basic strategy of proof is more or less the same as T. Shintani's 1976 paper On an explicit formula for class-1 “Whittaker functions” on GL$_n$ over $p$-adic fields, but there are much more complicated case-by-case arguments.
I have contacted the Tokyo library, perhaps they can make a scan of the handwritten text available.