Age of information (AoI), a metric to analyse data freshness, is of interest for time-critical applications in upcoming wireless networks. Besides, wireless power transfer (WPT) is also essential to prolong lifetime of a wireless sensor network. Thus, we study a flat-fading wireless powered cooperative network, wherein a source and a relay charge their finite-sized capacitors by harvesting energy from a remote power station. For two conventional decode-and-forward (DF) and amplify-and-forward (AF) relaying, average AoI and peak AoI (PAoI) are studied to show how the randomness of the data/power transfer channel and size of the capacitors affect the age metrics, and when utilizing a cooperative transmission is more beneficial than direct one. It is shown that, although the power-limited relay imposes more delay overhead, the age metrics are improved in some circumstances. Even for the DF scheme with more waiting time for signal processing, we may achieve lower average PAoI due to enhancing the end-to-end communications reliability. Furthermore, special cases of the proposed WPT cooperative system is discussed, e.g. cases of full-power nodes and one-shot successful data transmission. Finally, numerical results are provided to highlight effects of transmission power, size of the capacitors, and path loss exponent on the age metrics.