Learn Turkish: Ben yalnızım.

It is the "to be" suffix for "I" (meaning "I am"). Since yalnız (alone) ends in a consonant and its last vowel is an undotted ı, we use the matching suffix -ım based on 4-way vowel harmony rules.

Do I always need to include the word Ben in this sentence?

No, you don't. Because the -ım suffix already tells us who is speaking, you can simply say Yalnızım to mean "I am alone". Including Ben just adds a little bit of emphasis, as if saying "As for me, I am alone."

The lesson mentioned a -y- buffer for "I am", like -(y)ım. Why isn't it used here?

We only use the buffer letter y when the root word ends in a vowel, to prevent two vowels from crashing together (like in iyi-y-im). Because yalnız ends in a consonant (z), the -ım suffix attaches directly to the word without needing a buffer.