In this paper, we study the information freshness of short-packet communication in wireless networks, where a base station (BS) sends time-sensitive status updates to users via adaptive multiple access technology. To improve the Age of Information (AoI) performance of the network, the BS adaptively switches among non-orthogonal multiple access (NOMA), orthogonal multiple access (OMA) and cooperative NOMA with simultaneous wireless information and power transfer (SWIPT). Specifically, the BS carefully decides the appropriate multiple access technology and the corresponding power allocation according to the state of the network to optimize the expected weighted sum of AoI (EWSAoI) of the system. To this end, we first analyze the EWSAoI of these three multiple access technologies and propose an adaptive NOMA/OMA/cooperative-SWIPT-NOMA transmission scheme. In specific, we formulate a Markov Decision Process (MDP) problem and develop an optimal policy for the BS to decide whether to use NOMA, OMA or cooperative-SWIPT-NOMA for downlink transmission based on the current state of the network. We further prove the existence of optimal stationary and deterministic policy. Furthermore, to reduce the computation complexity, a suboptimal adaptive policy based on Lyapunov Optimization is also devised, which can achieve near optimal performance according to our simulation results. The extensive simulation results demonstrate the advantages of the proposed policies, which provide useful insights for practical system designs.