What is an antiunitary operator?

As Qmechanic noted, $T$ is antilinear (this is part of the definition of being antiunitary). Of course, $T^{-1}$ must be antilinear as well because $T$ is. Thus, for any vector in this Hilbert space $v$, $T^{-1}(iv)=-iT^{-1}(v)$. The $i$ pops out as a $-i$. Applying this to your equation, we easily have that $$ T^{-1}iT=-iT^{-1}T=-i. $$

If I correctly understood your misunderstanding, the answer is: operator is not always a matrix. Technically, action of time inversion operator contains complex conjugation. E.g., in spin up/spin down basis it is written as $-i\sigma_y\mathcal{K}$, where $\mathcal{K}$ is complex conjugation.