Translation:I was very tired yesterday.
I believe that's incorrect, although I'm also a learned. Based on jisho.org 疲れる (つかれる) is "to get tired", not "to tire" so the subject (grammatical subject, not the sentence topic marked with は) should be a person, not a day or an object. There is no person in this sentence so "I" is implied.
眠い・ねむい is an adjective closer to sleepy/drowsy.
Its verb form 眠る・ねむる means to sleep, to die, to be dormant
つかれて is a form of the verb 疲れる・つかれる - to get tired, to be worn out (as if being exhausted from work)
They're certainly similar but you can be physically/mentally tired without being sleepy