、！
I'd rather just give up than have him taken by another woman!
Literal
Other [attributive-の] woman [by-に] be-taken [rather-than-よりは], give-up settles [because-もん]!
A '~より(は) X' comparative of preference: X is preferable to being taken. 取られる is the passive of 取る — with に marking the agent ('taken by another woman'). 諦めつく is a less-common verbal form of 諦めがつく ('reach the point of accepting,' literally 'giving-up sticks') — you can come to terms with it. The sentence-final ~もん (colloquial for ~もの) adds a pouty, reason-giving 'because!' tone characteristic of young/feminine speech.