We consider the timelike version of warped anti-de Sitter space (WAdS), which corresponds to the three-dimensional section of the Godel solution of four-dimensional cosmological Einstein equations. This geometry presents closed timelike curves (CTCs), which are inherited from its four-dimensional embedding. In three dimensions, this type of solution can be supported without matter provided the graviton acquires mass. Here, among the different ways to consistently give mass to the graviton in three dimensions, we consider the parity-even model known as new massive gravity (NMG). In the bulk of timelike WAdS(3) space, we introduce defects that, from the three-dimensional point of view, represent spinning massive particlelike objects. For this type of source, we investigate the definition of quasilocal gravitational energy as seen from infinity, far beyond the region where the CTCs appear. We also consider the covariant formalism applied to NMG to compute the mass and the angular momentum of spinning particlelike defects and compare the result with the one obtained by means of the quasilocal stress tensor. We apply these methods to special limits in which the WAdS(3) solutions coincide with locally AdS(3) and locally AdS(2) x R spaces. Finally, we make some comments about the asymptotic symmetry algebra of asymptotically WAdS(3) spaces in NMG.