Ah finally, one that I can answer with primary sources from my personal research.

Ancient China generally refers to Shang to Han dynasty China, so while you learned about Han dynasty China, I will give an affirmative answer dating to Western Zhou dynasty, which predates the Han by nearly 800 years (about 1000 B.C.E.). In short, yes, some women, pointed out double standards. It seems that many (probably most) accepted it, with good reason.

You ask about common women's opinion, but in general for ancient patriarchal societies, this is tough to answer as the writers of the time were almost entirely men, and if there were women, they were usually elite. We can guess at common women's opinions sometimes, but that's basically just speculation. However, remarkably, for Ancient China, we actually do have some women's sentiment towards the double standards, and many scholars of Ancient China believe these sentiments come from common women. My source is the Shijing, or The Odes, a collection of ancient songs that would be recited orally before being written down in some form and edited around the Han. But for good reasons, we can assume some of the 'writers' of the Shijing were common women, as due to the oral transmission of the Shijing, some folk songs were mixed in. There are only a handful of poems, but taken as a whole, we can say affirmatively that some (including commoners) women pointed out the hypocrisy and double standards of the times.

One such poem complains of an unfaithful marriage, and seems to be from a common woman:

《氓 - Mang》or Simple Man (Poem 58):

You seemed a simple man, bearing cloth to trade for silk,

But you’d not come to trade for silk, you’d set your mind on me.

We said farewell at the River Qi ford, we walked you to Dun Hill.

It was not I who’d put it off, but you with no matchmaker.

“Please don’t be upset, we’ll fix an autumn date.”

I climbed a high wall to gaze toward Fu Pass where you lived,

And when I could not see Fu Pass, my tears fell flowing down.

And when I did see Fu Pass, how gay I laughed and talked!

Then when you threw the yarrow-stalks, they showed no baleful omens,

And so you came back with your cart, carrying off my dowry and me.

Before the mulberry sheds its leaves, how soft and glossy they are!

Oh dove, oh turtle-dove, don’t eat the mulberry fruit!

And oh, young girls, don’t take your pleasure with men!

A man may take his pleasure and still he may be free.

A girl who takes her pleasure, there’s no escape for her.

The mulberry leaves have fallen, all yellow and all sere.

And since I came to you, I’ve swallowed poverty three years.

The River Qi was in flood then, the bride cart curtains wet through
It is not I at fault , you have changed your path,

It is you who have broken faith , your favors cast this way and that.

Three years as your wife, and never slack in work.

Rising early, late to bed, never a moment, never at rest.
You found fault at first, and at last you grew rough.

My brothers disown me with chortles and laugh s .

All calm I think back , I alone console myself.

“ I shall grow old with you, ” I once vowed, but age makes me only resentful .

The Qi has its banks, the lakes have their shores ,

With hair looped and ribboned , how gaily you laughed, how solemnly swore to be true.

Never dreaming of change – of no change did I dream –

All ’s finished , alas, all is done!

- Robert Eno's Translation here: https://scholarworks.iu.edu/dspace/bitstream/handle/2022/23437/1.4-Poems-2010.pdf?sequence=2&isAllowed=y

It's quite a beautiful translation that shows, indeed, common women in Ancient China did see the double standards of the time. The Chinese character 氓 is an archaic way of saying commoner, so perhaps a better translation of the title might be commoner. With strict class endogamy in Western Zhou, her mention of his being a simple man would seem to indicate her being of a similar status. Further with mention of his unfaithfulness (the original says something to the meaning of: you have been double in your conduct), and reference to how she is unable to escape the marriage, this very clearly demonstrates that yes, common women at the time understood and knew the double standards of the time, and called it into hypocrisy.

Actually, your question gets a little dicey in its inception. For many Ancient cultures, this is a huge span of time and a bit too broad a question, but for China specifically, women's rights changed a lot across those centuries. The excellent and comprehensive treatment of women and gender in Ancient China, from Shang to Qin, Bret Hinsch's Women in Ancient China goes into it all, but basically, Western Zhou had expectations of constructive and equal marriages, and a majority of the Shijing's poems do mention women longing for their men away at war, though it may be male writers or authors for many of those (though not all). During Shang and Western Zhou, women possessed more rights, with there being powerful Warrior Queens such as Fu Hao who led soldiers to battle, and with women deciding land disputes on Bronzes dating back to the Late Western Zhou. As time went on, their domain became more and more restricted to the interior of the household. But this wasn't a total check on their power, and many women took near complete control on all matters within the home, as well as in their influence over kin, especially their sons. In Western Zhou, filial piety also reflected piety between mother and son, before that was eroded to father and son by Han. But even in Han, mothers exercised huge and near total de facto influence over their sons (Mark Edward Lewis's The Early Chinese Empires: Qin and Han), and there were many famous Empress Dowagers or Regents during even the Han who ruled 'behind the curtain.'

I don't know if many women had their thoughts on the double standards of Ancient Chinese societies recorded besides those in the Shijing, but based on the actions of the majority of women at the time, it seems they had a keen understanding of their place in society, and knew how to exploit it.

Further, we have some circumstantial evidence from Eastern Han. The famed noblewoman, writer, and historian Ban Zhao wrote the book Lessons for Women. The book is famous in history for being written by a woman about how women (unfortunately) should be subservient, (it describes how a baby girl must be placed on the floor in subservience to the men in their lives, for example) but Ban Zhao is nobility, so we cannot generalize her beliefs entirely to all Han Chinese women, and the existence of a book would seem to indicate a fair number of women were doing the exact opposite of the prescriptions in that book.