I have an unstemmed kiddush cup with Hebrew inscriptions on the inside. I am told that it is from Rabbi Hashrash and refers to the 4 rivers of Eden. Do you know the story behind this?
Yes I do actually. It is a segulah, supposedly a surviving part of the book of refuot and segulot that was hidden by King Hezekiah. If made according the specifics brought down by the Rashash, and available online in the siddur of the 5th Rosh Yeshiva of Beit El Rav Raphael Yedida Abulafia(it runs on for several pages), the letters would be hand engraved with specific Kabbalistic intentions(or Kavvanot) and prayers while being engraved.
Today there are two versions of this cup commercially available, one is with hand engraved letters, and one is factory produced with laser engraved letters. According to Rav Benayahu Shmueli, in his sefer Pidyon Nefesh p. 263(Second Edition published 2011/5771), the latter is inferior, but still a valid segulah. As well he brings there(p. 264) a prayer to be said before usage as well as various other usages for the names and how that can be accomplished.
Quoting directly from the text:
הנה כי אלו האותיות הם המפתחות שערי גן עדן פותחים שערי הרחמים כולם ומורירידים אותם... ובכל יום שתזכור אותם כאשר עוסקים בתורה, או בלילה על מיטתך, וגם פסוק ויהי נועם כולו, ופסוק ואני תפילתי לך ה' עת רצון, תהיה מוצלח כל אותו יום, ובכל אשר תפנה תשכיל ותצליח, ותכבוש הצד השמאל כולו תחת ידך, ואין מי שיעמוד לפניך
These letters are the keys to the gates of heaven open gates of mercy and draw them all down ... And every day that you remember them when occupied the Torah, or night on your bed, and the entire verse Vahi Noam, and the verse Vani Tafilati, you will be successful in everything that day, and all undertakings will prosper and succeed, and you will subjugate the other side, and no one to stand before you.