Learn Finnish: Hän tuli juuri.

You attach the -kin suffix to the word you want to emphasize. To say 'He also' or 'He too', you attach it to hän, making the sentence Hänkin tuli juuri.

What does it mean if I attach '-kin' to the verb instead, like 'Hän tulikin juuri'?

Attaching -kin to a verb often translates to 'actually' or expresses that an action was slightly surprising but did happen. Hän tulikin juuri would mean 'Actually, he just came' or 'He did just come after all'.

Why is the past tense of 'tulla' just 'tuli'?

Tulla (to come) is a type 3 verb. To find the stem, you remove the -la, leaving tul-. To form the past tense, you add the past tense marker -i-. Because the 'hän' form doesn't take a personal ending in the past tense, it remains simply tuli.

I learned before that 'juuri' means 'root'. Is it the same word here?

Yes! Juuri literally means a plant 'root', but it is also very commonly used as an adverb meaning 'just', 'exactly', or 'right now'. You can usually tell which meaning it is by whether it's being used as a noun or an adverb describing time.