As the person giving the gift to someone, can I pay the gift tax, too?

Yes, you can. However, if you do so, you must take account of the fact that also the tax is a gift, so gift tax must be paid on it.

Accordingly, if you give an apartment to someone as a gift − or if you give some other property − and you want to make it so that the recipient of your gift will not have to pay gift tax, there’s nothing to prevent you from donating the necessary sum of money to cover the gift tax, too. The base for gift taxes is the total gift value — the amount that includes the apartment and the cash gift you give in order to pay for the gift tax.

Another important point is that the base also includes all the other gifts you may have given during the past 3 years. If gift tax was paid on those already, the Tax Administration will take it into account: you will only have to pay the part that your previous gift-tax payments do not cover.