Estate and inherited property

Client is a 10% beneficiary of a property which is still in an estate. If the estate sells the property, the tax is to be reported in the Estate tax return and it will have nothing to do with the client's personal tax return (even though he is one of the beneficiaries). Is it correct?

Maybe. If the gain is reported on a final 1041 it will pass through to the beneficiaries on their K-1s.

Even if it is not a final estate tax return, would distributions to the beneficiary in the current year be passed through and taxed in the beneficiary’s personal tax return?

Income distributions would, but capital gain is generally corpus and doesn't pass through.

Just to make sure I have understood correctly.

Let’s say an estate had income of $10,000 capital gain and $3,000 rental income in 2020, and the total of $13,000 was distributed to the beneficiary in 2020 too. The $3,000 rental income would be passed through and taxed in beneficiary’s personal tax return. But the $10,000 capital gain would still be taxed in the Estate’s tax return. Is that right?

That's generally how it works.