This method may or may not work, and will probably work better if one be a native speaker of some form of British English, but try saying “syi” as in the onset of “suit”, when it be pronounced as “syoot” instead.
The British English realization of “sy”, as distinct from “sh” is actually fairly close, and also shares the same underlying phonemic sequence with the Japanese sound. Remember that “sh” is not considered a single sound in Japanese, it is rather analysed as a sequence of two sounds that simply assimilate into each other in pronunciation.
Speakers of dialects of English that lack “sy” and pronounce “suit” identical to “soot” might find it difficult to pronounce “sy”, however — one's mileage may vary.
P.S.: I did a spectogram of British English: “soot”, “suit” and “shoot” as well as a Japanese “しゅ”:
The part with the dark bands at the bottom is where the vowel starts, before that is the sibilant-like sound in each language where the dark band is at the top. As you can see; in English “s” the dark band is at the top, and in English “sh”, it is in the middle, whereas in the Japanese sound, it is in between both, as well as in the English “sy” in “suit”.
One thing of interest is that English “s” and “sh” are constant, whereas in English “sy”, the dark band moves noticeably. In the Japanese sound the band also moves, but to a lesser degree, indicative of that in both languages these are actually two sounds, not one, that to some extent transition into each other.
In English “syoo” however, the vowel seems to have elements of a diphthong, as one can see a dark band flowing down, and transitions from something more “i” like to something more “u” like. This is præsent in the Japanese to a far lesser extend and can probably be omitted, but the Japanese vowel in “しゅ” does still seem to slow some slight progression from a i-like vowel to a u-like vowel. Whereas in English “soot” and “shoot”, the u-like vowel after it is nigh completely constant.
I also cut out the sounds used for comparison by ear, in the same order:
I also now did a spectogram of one speaker's rendition of the “し” in “知らない”, to satisfy my own curiosity more than anything:
This also seems to indicate a mild transition during the fricative, indicating that the tongue position isn't constant, but to a far lesser extend than in “しゅ” and certainly the English sample of “suit”. There are also more dark bands in the middle at the end than at the start here.
So lessons to be learned from this:
the Japanese “し” and “しゅ” sound seems to be somewhat in between the English “s” and “sh” sounds, probably closest to “sy”, but a bit more in the “sh” direction than “sy” normally is in English.
The English “syoo” sound shows a far more prominent set of transitions than the Japanese “しゅ” sound, which shows more than the Japanese “し” sound, but both still show transition whereas English “soo” and “shoo” are constant.