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I have been banging my head against a wall trying to understand these time conversions and the relation between all of these astronomical measurements.enter image description here

This is the question I am currently working on, and I am honestly lost on where to get started.

I have heard that Local Sidereal time is just equal to the RA of a star at the meridian, so from that logic (a) should be the RA of 51 Eridani (about 4hr 37min). But why does this part of the question give the longitude of the observatory? I feel like I am completely missing something, and to be honest, I do not feel like I understand what a sidereal time actually is.

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You are correct; the right ascension of an object equals the local sidereal time of its meridian transit. The geographic longitude of the observatory is only relevant to part b.

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