The first time iI learned about the epidemic in Wuhan was on 5th, Jan 2022, 16 days ahead of the nationwide lockdown over the whole Sspring Hholiday.

Formats for dates are usually like this
- the 5th of January 2022
- 5 Jan 2022

"the" because it's a specific epidemic

the spring holiday is not really a name of a holiday, I presume, so it's just lowercase

I remembered it was onduring an optional course ofn financial laws wheren my teacher told us that he just returned from an epidemic zone — Wuhan cCity, Hubei pProvince, which incurred our laughter because we didn't believe there would be epidemic in this era when mordern medicine is so developed.

laugh is the verb not the noun

I put "when" instead of "where" because your course is more like the time it took place, rather than the place

2Two weeks later, a 3-three months lockdown began, which immediately disturbed all my plans.

For numbers under twenty, usually they are written in words (it looks better)

It's always all + plural noun

We had just broken up before Sthe spring Hholidays because she wanted to check all my messages on my social accounts, which ended up in a fierce quarrel.

That's rough :(
Again with the capitalisation of spring holiday, if it's actually the name of the holiday then you are right to use capitals

I used "had broken up" to imply that this is an event in the past that happened before the pandemic

I intended to make up this relationship after backreturning to university, but suddenly found myself left in a situation where all students except graduates were banned from backingcoming back to school untill next term.

till is an informal abbrieviation

TilUntil one day iI met a guy, the same age withas me, on a online talking room, who shared his experience as an volunteer in Wuhan — the center of the epidemic.

He responsded to the nation's call and set off from his hometown to Wuhan byon an empty train, deserted because people were scared of being infected through face-to-face contact due to which discard train as approach for trip.

"an" not "a" in front of words starting with a vowel

I didn't really understand the last part of the sentence so I changed it as above

The first time iI learned about epidemic in Wuhan was on 5th, Jan 2022, 16 days ahead of the nationwide lockdown over the whole Spring Holiday.

I remembered it was onduring an optional course of financial laws where my teacher told us that he just returned from an epidemic zone — Wuhan city, Hubei province, which incurred ourmade us laugh because we didn't believe there would be epidemic in this era when mordern medicine is so developed.

"made us laugh" is more common

2 weeks later, a 3-months lockdown began, which immediately disturbedupset all my plans.

We just broke up before Spring Holiday because she wanted to check all messages on my social accounts, which ended up in a fierce quarrel.

I intended to make up this relationship after back to university, but suddenly found myself left in a situation where students except graduates were banned from backreturning to school untill next term.

There is always a space after a comma:
"I was going shopping, but then I took a nap instead."

Till oOne day iI met a guy, same age with me, on a online talking room, who shared his experience as an volunteer in Wuhan — the center of epidemic.

Till doesn't technically exist, though sometimes in casual writing or speech it can be 'till
"She can complain 'till the cows come home"
In writing it should always be "until"

First person pronoun is always capitalized, "I" instead of "i"

He responsded to the nation's call and set off from his hometown to Wuhan byin an empty train, because people were scared of being infected through face-to-face contact due to which discard train as approach for trip.

I think the second part of the sentence can be implied, though you could also say something like:
"...because people were scared of being infected through face-to-face contact they cancelled their travel plans."