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If the amendment specified that it did apply to the sitting president, then it's legal. Nothing surpasses the Constitution and it's amendments.
I have mixed feelings about the proposed bill only working for newly elected senators. It'll be easier to pass if we allow an exception, but then we'll have a situation for the next couple decades where we have VERY senior senators. 
Edit: 
Maybe the wording can say:   people can only be elected 2x beginning n 2023.
They say this stuff knowing full well it’ll never pass because they all want to keep the grift running and line their pockets from donors. It’s all just garbage talk to get people excited for them. >

Term limits for presidents is a good idea. The power difference between a senator and president is stark and the more time the president has in office the more likely he could become a dictator if he wanted too. FDR had some pretty authoritarian tendencies. I think a long term limit like 18 years for Senators and 20 for house members is good. It prevents the Mitch McConnells from sticking around and still let's experience be there >

I agree. Two terms isn’t enough for a Senator. Maybe four, the out the door. >

Ted Cruz only cares about Ted Cruz. He wants term limits for everyone else but that not in his best interest so he's just going to keep running for reelection regardless. He's a self centered hypocrite and he's aware of that and he doesn't care >

In his defence (I never thought I would say this) it's very much illegal to retroactively limit the rights of elected officials in most places.
I'm not in the US, but I am pretty sure that retroactive laws in general  are unconstitutional. >

It would be wrong to set a law that removed any senators that have served for more than two terms. It would not be an issue to pass a law to make them ineligible to run for another one. >

The constitutional 2 term limit for presidents did not apply to the  Harry S. Truman who was the incumbent while it was ratified. I would assume that the same thing would apply here. >

It didn’t apply to Truman because it didn’t apply to Truman.

But this Article shall not apply to any person holding the office of President when this Article was proposed by the Congress, and shall not prevent any person who may be holding the office of President, or acting as President, during the term within which this Article becomes operative from holding the office of President or acting as President during the remainder of such term. >

Sounds like legalese to make sure that they don't retroactively limit the rights of elected officials, which as I said before, is probably not legal.