Nearly all visa runs to Japan take place in Osaka. The consulate is no longer able to process a visa the same day. You must bring in your paperwork before 11 a.m. on a weekday and then you may pick up the visa the next day after 1:30 pm. The hours on the site state that the consulate is closed during the noon hours, but exceptions are made for E-2 visa applicants as they are aware that the first plane from Seoul to Osaka leaves around 10:00 am and that it is logistically impossible to arrive before noon. If the E-2 visa applicant arrives before 1:30, the visa will be ready at 4:00(4:30?) pm.
Usually your company will give you something called a Certificate of Confirmation of Visa Issuance. Obviously you need to take this with you to Japan. And make sure that you bring a passport-size photo of yourself. I beleive there is a photograph place in or near the Korean Consulate in Osaka. When you get to the consulate (indentifiable by 2 huge Korean flags on either side of the door) you will need to take a number from the little machine and wait for it to be called. While waiting, fill out one of the Visa Application forms. Even though you have the Confirmation form you still need this form. Trust me, I got sent to the back of the line for not filling one out and had to wait 2 extra hours. Your waiting time will depend on how many people are applying for visas (which takes about 10 min per person) and how many people are coming back to pick up their visas (which takes about 1 minute per person) When your number is called they'll check your form and tell you how much you have to pay (around 5,000 yen (50,000 won). and they only take yen. no wons are accepted!). After you pay you get a strip of paper and (probably) instructions to return the next day. the next day you just have to wait in line again and pick up your passport with the shiny new visa pasted in it.
This is a hotel for about 1800 yen per night (no joke. about 18 bucks a night. in Japan!) in the Umeda part of Osaka, which is fairly close to the consulate. It's (incredibly long) name is 紀州(kisyu)鉄道(testudou)大阪(osaka)梅田(umeda)ホテル(hotel). It's website is http://www.kitetsu-hotel.jp/osaka/