The Cyclops is a dangerous foe to melee toe-to-toe. He does a lot of damage and will drain your experience level with The Staff. The Cyclops respects Elbereth. Simply burn Elbereth on the upstair with a wand of lightning or fire. He will teleport next to you but not attack, allowing you to dispatch him at your leisure. You could also engrave in the dust multiple times but this is somewhat riskier. If he doesn't teleport over to you then he hasn't noticed you (probably because you are invisible). You'll have to catch his attention somehow - either teleport over to him and then back or make some noise.
The Cyclops' guaranteed wand of lightning poses a strategic dilemma. The standard defense against lightning is reflection, which protects your equipment, whereas shock resistance alone does not. However, reflecting the Cyclops' lightning back at him will blind him, causing him to ignore Elbereth. If you're unwilling to go without reflection, note that casting the spell of extra healing at the Cyclops will unblind him. After at most eight zaps, his wand will be out of charges, and you can finish him off from the safety of your E-square.
If you choose not to go down the Elbereth route, you should note that The Cyclops has 0 MR and no resistances other than stoning. Zapping him with death, polymorph or even repeated application of sleep will make dealing with him much easier.
The Cyclops starts surrounded by several dragons, and if you approach them both invisible and generating conflict there is a good chance the dragons will hurt the Cyclops enough that he teleports to the upstairs without picking up The Staff. If you grab it yourself he is considerably easier to defeat.
Note that Cyclops, like any giant, may be eaten for a strength bump.
And after he had milked his cattle swiftly,
he again took hold of two of my men
and had them as his supper.
Then I went, with a tub of red wine,
to stand before the Cyclops, saying:
"A drop of wine after all this human meat,
so you can taste the delicious wine
that is stored in our ship, Cyclops."
He took the tub and emptied it.
He appreciated the priceless wine that much
that he promptly asked me for a second tub.
"Give it", he said, "and give me your name as well".
Thrice I filled the tub,
and after the wine had clouded his mind,
I said to him, in a tone as sweet as honey:
"You have asked my name, Cyclops? Well,
my name is very well known. I'll give it to you,
if you give me the gift you promised me as a guest.
My name is Nobody. All call me thus:
my father and my mother and my friends."
Ruthlessly he answered to this:
"Nobody, I will eat you last of all;
your host of friends will completely precede you.
That will be my present to you, my friend."
And after these words he fell down backwards,
restrained by the all-restrainer Hupnos.
His monstrous neck slid into the dust;
the red wine squirted from his throat;
the drunk vomited lumps of human flesh.