The "ha" sound is like in hash, but it's a distinct syllable on it's own, without the "sh". The second syllable is more like "she". The "ha" and "shi" have the same weight, so the whole word is 2 syllables and 2 morae, unlike either of the two examples you gave (which are 3 morae each.)

It usually means "bridge" (橋) in the name Hashimoto, but can also mean "chopsticks" (箸) or "edge" (端), with a slightly different pitch (largely indistinguishable for non-speakers)