See the Sefer Aim Habonim Semaicho Perek 3 in great detail what comes first Messiah or the Building of the Bes Hamikdosh.
Also from the same author Rav Yisochor Teichtal in Mishneh Sochir in Parshes Beshalach מקדש ה כוננו ידיך in great detail discussing this topic.