My teacups, part 3.

It's not hard to find a "zZojirush" teacup with a filter screen.

Capitalise company and brand names
I think you can just say filter without 'screen', I understand what you mean without it

This time, I stillagain chose a black one, but this one was not 360 ml, but 340 ml.

Did you purchase a black one before? I hope this is what you meant, you need to replace 'still' with 'again'

Its filter screen iwas not made of stainless steel, but of plastic components.

You were discussing the same item in the past tense in the last sentence so I would stick with the past tense again

Before I made the tea, I would wash the tea first.¶

Either:
Before I made the tea, I would wash the tea first.
Before I would make the tea, I would wash the tea first.
Articles again. You said 'tea' then 'the tea' in the same sentence

Because of the filter screen, the ability to washing tea inside the cup was very easy.

Reading this it took me a couple of times to understand it. It doesn't flow so easily so I added something to split up the clauses better

After I washed the tea, I preferred taking the filter screen away.

SomehowFor some reason, I didn't like drinking tea with the filter screen oin.

I think maybe I have a kindsome sort of obsession.

After I used this cup for a few days, I learned some information which said that the insulated cups were not suitable for making tea and even harmful sometimes.

People think that a glass cup is the best cup for making tea, so I want to buy a glass teacup again.

Indefinite articles are needed here