Let's say the house is worth €500,000 at the date the transfer happens, and there is €160,000 outstanding on the mortgage.
You are getting a gift of (500k/2 =) 250k, subject to taking on a liablity for (160k/2 =) 80k, so the net value of the gift is (250k - 80k =) €170,000.
Because you're unmarried your ex is in category C for CAT purposes; your lifetime threshold for gifts and inheritances from people in category C is €20,000. Assuming you have received no other chargeable gifts or inheritance from people in category C, you can offset that, so the amount charged to CAT is (€170k - €20k =) €150k. CAT at 33% is €50,000.
You could possibly reduce this if you could argue that there were other assets that you were giving, or giving your share of, to your ex, and the value of these should be offset, but your post suggests that's not the case.
The other avenue that might be worth exploring, if the facts support it, is arguing thay you have a greater than 50% interest in the house already, because you paid more than half of the deposit and/or the mortgage payments, and CAT should be calculated only on the gift of so much of your ex's interest as you can't claim on the basis of your contribution to the purchase.
I assume that the house was the principal residence of yourself and your ex, so the question of capital gains tax on disposal will not arise. (And, if it did arise, any CGT would be your ex's problem, not yours, because he is the one disposing of an interest in the house.) Similarly, I assume that it will continue to be your principal residence, so we do not need to worry about your liability for CGT if and when you eventually dispose of it.