Two versions:  a corrected version and an edited version.

Corrected Version

Waiting for an E-mail

Note -

"... if I won't be able to attend classes there, I have to enroll at language classes at college." is acceptable but somewhat non-standard.  A classic type 1 conditional is "... if I can't attend (present tense for a future condition) classes there, I will have to (future tense for a future result) enroll at college."

More information about conditionals here:

https://www.ef.com/ca/english-resources/english-grammar/conditional/

Edited Version

Waiting for an E-mail

On the last workday before the New Year's holiday, I called the Japanese school where I used to have classes. I wanted them to give me infomation regarding the time and prices of the regular and the summer their January courses. The son of the owners answered (since it's a small school) and told me (that) he didn't have the table information at hand with him at the moment, so he said , but (that) he would send it to me in an e-mail. I'm worried because the e-mail hasn't arrived yet, and I need the information as soon as possible, not only because classes there start on January 8th, but also because, if I can't attend their classes that start on January 8th there, I will have to enroll immediately in language classes at a different school.

Waiting for an E-mail

On the last workday before the New Year's holiday, I called the Japanese school where I used to have classes. I wanted infomation regarding  their January courses. The son of the owners told me (that) he didn't have the information at hand, but (that) he would send it to me in an e-mail. I'm worried because the e-mail hasn't arrived yet, and if I can't attend their classes that start on January 8th, I will have to enroll immediately in language classes at a different school.

Waiting for an e-mail:

Waiting for An E-mail

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You're pretty good at English. I only had minor corrections. If you need someone to speak English with for practice, you can sign up for lesson in my teacher profile.

Waiting for an E-mail

On the last day before the New Year's holiday, I called the Japanese school where I used to have classes. I wanted them to give me some information regarding the time and the prices of both the regular, and the summer, courses. The son of the owners answered (since it's a small school) and told me that he didn't have the class schedules with him at the moment, but he said he would send me an e-mail with it later on.

I'm now worried because that e-mail hasn't arrived yet! I honestly need that information as soon as possible not only because classes there will start on January 8th, but also because I might need to enroll in language classes at a college if I am unable to attend classes at the Japanese school. And, most likely, their deadline to register is closing soon also!

Hi Julia. Good job! Whatever changes that I’ve made were done to merely allow your sentences to sound more natural to my native ears. I hope that his help you. J.

“On the last day”—This date is somewhat unclear. If you meant the 31^st, which is logically the last day of December, then simply use that, or “This past Monday….” However, if you meant another time (especially, the last time the school was open…), that may have been weeks ago. Being vague only creates confusion in the reader’s mind.

“the time and the prices”—students often forget to use the second “the,” but it reads better and adds balance to your sentence (if it is used before “time,” why shouldn’t be used before “prices”?).

"enroll at”—you enroll in classes, so review the differences between “at” and “in.”

Capitalization of Titles-I use Case Title [i.e., I capitalize every word except articles, conjunctions and prepositions], but regardless, your titles need to follow some capitalization style. Find one that you like and use it consistently.

The son of the owners”—is fine, and denotes that the owners are two people (his parents). Someone else suggested that the alternative would be “The owner’s son,” which is singular. It should be “The owners’ son” as the possessive of a plural noun (owners) is formed by adding only an apostrophe when the noun ends in s, and by adding both an apostrophe and s when it ends in a letter other than s. 

Waiting for a e-mail