What is the subject of the first sentence?what does "so that" means? What is the role of two last phrase, could you paraphrase it for me

Just as painted designs on Greek pots may seem today to be purely decorative, whereas in fact they were carefully and precisely worked out so that at the time, their meaning was clear, so it is with Chinese pots.

As deadrat pointed out, punctuation could be better:

Just as painted designs on Greek pots may seem today to be purely decorative, whereas in fact they were carefully and precisely worked out so that at the time their meaning was clear, so it is with Chinese pots.

Consider it this way:

[Just as painted designs on Greek pots may seem today to be purely decorative, whereas in fact they were carefully and precisely worked out so that at the time their meaning was clear], [so it is with Chinese pots.]

What you have here is a "Just as... so it is..." construction. Here you can see an example of this from the Corinthians:

a. "Just as a body has many parts, so it is with Christ" [abridged for clarity]

"As" is a synonym for "like". The author is comparing two situations and stating that they are somehow the same:

A body has many parts, and Christ does too.

Roughly, whatever is said about the first item, it also applies to the second. In (a), the first item is "a body" and the second is "Christ". In your sentence, the first item is "Greek pots" (or the designs on Greek pots), and the second is "Chinese pots" (or the designs on them).

"So that" is also a fixed expression meaning "in order to" or "that way". Consider:

b. I'll do it, so that you don't have to (= I'll do it. That way, you won't have to.)

So the author is saying that Greek pots were carefully painted in order to make their meaning clear. "At that time" only means "then" or "back then". "Whereas" is a conjunction expressing opposition. It can be replaced with "but". So you have:

Painted designs on Greek pots may seem just decorative today, but they were carefully worked out so that (at the time) their meaning was clear. The same is true with Chinese pots (their designs were also meaningful in their time, although they may look merely decorative nowadays).

You basically have two clauses, the first one with three clauses within it:

A:[(₁Just as painted designs on Greek pots may seem today to be purely decorative₁), (₂whereas in fact they were carefully and precisely worked out₂) (₃so that at the time their meaning was clear₃)], B:[so it is with Chinese pots.]

• The subject of a is the Noun Phrase "painted designs on Greek pots", being "designs" the most important word (or head).
• The subject of 1 is the same as in a ("painted designs on Greek pots").
• The subject of 2 is the pronoun "they", whose antecedent is "painted designs on Greek pots".
• The subject of 3 is the Noun Phrase "their meaning", whose head is "meaning". "Their" is a possessive determiner referring to "designs".
• The subject of b is the pronoun "it" (whose antecedent can be considered a).