How much residence tax to pay?

My employer just sent me an email detailing how much residence tax I need to pay before leaving Japan. And it is humongous. If I am leaving say on or before 20th June (can be adjusted), how much tax I need to pay?
My yearly income ~ 5.4 million yen
P.S. I have been living here for around 5.5 years

It is difficult to know exactly without knowing the details of you situation, but if you are enrolled into employees' social insurance and have no other income, I estimate that you have to pay around 23,000 yen per month, which translates to an annual amount of around 275,000 yen.

Why have you not been getting bills from your local city hall for residence tax? Surely it would be similar this year to previous years rather than suddenly a very large amount.

The taxes you pay are based on last year's income. Since you are leaving in June, I assume they calculated this year's taxes too. So it's about 1.5 times normal.

Local taxes are paid with a delay of one year. As a result, foreigners still have to pay an entire year's amount after quitting the job. Also, the timing of the OP's quitting the job is in a way that his employer just received a bill for the OP's entire year of local tax payments. So, the employer has the duty to pay it. If the OP had left the company a little bit earlier, the OP rather than the employer would have received the bill.

I am always surprised about how many residents are not properly informed about their tax duties, but have to admit that the 1-year delay of local tax payments is hugely confusing, especially for temporary residents.

After we leave job, the resident tax is calculated. You will get the bills which you need to pay in the convenience store. Let me tell you my figure. When I had quit my job and yearly income was around 500man per year, I had got resident tax around 16 man which I need to pay in 4 installments.

You will get the bills which you need to pay in the convenience store

No, in this case, the employer got the bill. I think there are a few options on how to deal with it:
a) you pay the tax to the employer who submits it to the government
b) the employer withholds the amount from your salary and submits it to the government

Also, I think it may be possible for the employer to withhold the monthly payments only until you quit, and the rest of the payments would then be billed to you personally. But since you leave Japan soon, you won't be billed until after you leave, so you would have to find a way to receive the bill and pay later, which can be a challenge. So, it is probably be best to go with option a) or b) listed above.