The ``Lewis-Riesenfeld phases'' which play a crucial role in constructing shortcuts to adiabaticity may be a resource for the implementation of quantum phase gates. By combining ``Lewis-Riesenfeld invariants'' with ``quantum Zeno dynamics,'' we propose an effective scheme of rapidly implementing $\ensuremath{\pi}$ phase gates via constructing shortcuts to adiabatic passage in a two-distant-atom-cavity system. The influence of various decoherence processes such as spontaneous emission and photon loss on the fidelity is discussed. It is noted that this scheme is insensitive to both error sources. Additionally, a creation of
N-atom cluster states is put forward as a typical example of the applications of the fast and noise-resistant phase gates. The study results show that the shortcuts idea is not only applicable in other logic gates with different systems, but also propagable for many quantum information tasks.