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We seem to be talking about three different circular slide rules here: The Weems & Plath Celestial Slide Rule; the Air Force X-1 circular rule see http://fer3.com/arc/m2.aspx/Weems-Plath-Celestial-slide-rule-instructions-EdPopko-sep-2016-g36452 ; and the USPS LHA Aries Computer.  The last two would appear to be quite useful devices. The amber, blue, and white device labelled ‘Weems and Plath Celestial Slide Rule’ which was the subject of the original question in this ‘string’ is a much simpler device.  There is no date or hours scale, at least on the front.  There might be something on the back. Which is why I asked for further details. 
My worked-out instructions are for the front face of the Weems and Plath Celestial Slide Rule requested by Ed J.  Upon rereading, I’ve made two amendments in italics below to make to make them read more clearly.   

Incidentally, the 70KB picture of the W&P device provided in Ed J’s link has suddenly disappeared and been replaced by a holiday advert.  Fortunately, I kept a copy of the photo, also in a blurred 70KB, which I can send on privately if requested.  I have no photo credit to give or permission to publish here. DaveP [EDIT: fixed in original post]

To obtain LHA Sun (or Planet):  Align GHA Sun on the white scale with zero on the outer amber longitude scale.  Move cursor to the observer’s longitude on the outer amber scale and read off LHA Sun (or Planet) from the white scale.

To obtain LHA Star:  Align GHA Aries on the white scale with zero on the outer amber longitude scale.  Move cursor to the observer’s longitude on the outer amber scale and read off LHA Aries from the white scale.  Being careful not to move the cursor, slide zero on the blue SHA scale under the cursor.  Move the cursor over SHA Star on the blue scale and read off LHA Star from the white scale.  With practice the user might find an easier order to do this.