Along with the pp saying be careful not to touch the skin around the cold sore because you'll spread it, please can I add: whatever you do, do not touch your eyes if you've touched your cold sore. If you transfer it to your eye, the one on your lip will be the least of your worries.
Cold sore ulcers in your eye can damage your eyesight (even causing loss of sight in bad cases), and even at best will take weeks and weeks of 5 times a day acyclovir ointment, and then regular (every few weeks at first, then twice a year for a couple of years, then annually for pretty much ever) eye check ups with dye and specialist equipment to track the level of healing. And then there's the pain. The pain of an eye ulcer is like nothing you've ever experienced.
Can you tell I've had one in my eye? Luckily I've never triggered another one in my eye, but it took several years for my eye to settle, and I'll never be able to wear contact lenses. The scarring covers almost 20% of my cornea. I was very lucky not to lose my sight, and the specialist treating me said he'd seen grown men with smaller ulcers than mine literally rolling on the floor in pain.
OP, be aware too, going forward, that sunlight can also be a trigger - so sunblock on your lip/between your lip and your nose. (I wear sunglasses all year round to protect my eyes) And keep a supply of cold sore patches in so you're ready if another one ever takes you by surprise.
Good luck!