Can I revive Gust Monk by empowering Matriarch without having to spend the extra energy?

Yes — you can play Gust Monk off Tail-Cloaked Matriarch's Empower trigger without paying the extra [1].

"Play it to your base, ignoring its cost" only zeroes the base Energy and Power costs (Rule 356.1.b). Gust Monk's "[1] as an additional cost" is an optional additional cost — "may pay" — so you can simply decline it (Rule 356.2.b). He still qualifies as a target since his printed cost is 2 Energy / 0 Power (within Matriarch's 3 / 1 limit, per FAQ #6562's printed-cost precedent).

The only catch: if you skip the [1], you won't get the "banish a card from any trash to give Assault 2" effect — that only happens if you actually paid the additional cost. If you'd rather have that effect, you may choose to pay the [1] even while the base cost is ignored.