Learn Portuguese (Portugal): Tenho sono agora.

In Portuguese, you express being sleepy by saying you "have sleep." You use the verb ter (to have) plus the noun sono (sleep). This is very common for physical states in Portuguese, just like how you "have" hunger (ter fome) or "have" thirst (ter sede).

Where is the word for "I" in this sentence?

The word eu (I) is left out because the verb form tenho already tells us exactly who the subject is. In Portuguese, it is very common to drop the subject pronouns (I, you, we, etc.) when the verb ending makes it obvious.

Is sono the verb for sleeping?

No, sono is a noun that means "sleep" or "sleepiness". The actual verb for "to sleep" is dormir. So this sentence literally translates to "I have sleepiness."

Can I use the verb estar (to be) to say I am sleepy?

Yes, but you have to word it differently. You can say Estou com sono, which literally means "I am with sleep." Both Tenho sono and Estou com sono are very common and correct ways to say you are sleepy.