Mifare is a protocol that does cryptographic exchange with a secret that you cannot read. Basically, a secret question and answer game, with an uncloneable secret inside the chip. It can't be copied.
Unless it's cryptographically unsound.
Mifare classic is cryptographically unsound and has been broken (video), so that tags could be forged. However, that is way more involved than just hooking up some device or flashing something on an arduino.
So, at your apparent level of involvement with this: no. Instead of duplicating an ID card, you'll have to add another authorized ID card to whatever system you're dealing with.
Also: from your description it's not clear whether this is Mifare classic or one of the more recent versions, which are safe and can not be cloned at all, to the knowledge of the public.
If your system is using Mifare classic, urge the operator to upgrade, unless the risk / cost tradeoff doesn't support that.