Not well enough to justify the 2003 war.
(amongst other things, there is a high chance Iraq would end up in a similar place today without it)
It is insane and or dishonest to claim that Saddam Hussein would have fallen when Bashar Al Assad (who looks like a hippy next to Saddam) clung on. Oppose it or don’t, but don’t pretend this is true.
Assad’s regime is a shell of its former self, is more beholden to Iran and Russia than ever, and doesn’t even control all of Syria.
And Russia did have a chance to help Saddam when it “came to it”—between 9/11 and the US-led invasion. Why didn’t they?
Syria has much more strategic significance to Russia than Iraq did at the time. There's nothing of value for Russia to defend, if anything like Iran they probably couldn't believe their luck when the US took out an unreliable customer for them and handed it to the Shia militia backed by Iran.
It's worth remembering that what seemed to tip the scales for Putin opposing anything the West was proposing in terms of revising the Middle East and beyond : The botched Libyan humanitarian intervention that resulted in mission creep and overreach. The video of Gaddafi's corpse probably incentivized a lot of the ramping up of the backing of despots. Iraq was obviously a precursor to this but around that time Putin's doctrine still wasn't oppose the West in all theaters they intervene in.