Putin works with Israel when it suits Russian interests, and against them when it doesn't. They pretty much always vote against Israel at the UN, and they have a long history of supporting Israel's enemies.
Russia has been talking about Iran reducing it's presence in Syria now that ISIS is all but defeated, but Iran wants to keep a large, long term/permanent presence. Russia wants a stable Syria for it's military bases and pipelines, and knows Iran and Hezbollah attacking Israel from their Syrian positions works against their goals.
Getting Iran out also contributes to getting the US out, leaving Russia as the only significant foreign influence in Syria, a rare case where Russian, American and Israeli interests align.
Lots of ethnic Russians in Israel, and they generally don’t care for Muzzies, so that changes things.
He might also consider that sponsoring Arab wars against Israel is a losing proposition. If it’s war, they win and Russian engineering and assistance humiliates Russia. If it’s just sniping and skirmishing, we’re not going anywhere, so why not deal with us instead. We don’t do crushing defeats due to bad training and morale.