Temporal networks, whose links are activated or deactivated over time, are used to represent complex systems such as social interactions or collaborations occurring at specific times. Such networks facilitate the spread of information and epidemics. The average number of nodes infected via a spreading process on a network starting from a single seed node over a given period is called the influence of that node. In this paper, we address the question of how to utilize the partially observed temporal network (local and of short duration) around each node, to estimate the ranking of nodes in spreading influence on the full network over a long period. This is essential for target marketing and epidemic/misinformation mitigation where only partial network information is possibly accessible. This would also enable us to understand which network properties of a node, observed locally and shortly after the start of the spreading process, determine its influence. We systematically propose a set of nodal centrality metrics based on partial temporal network information, encoding diverse properties of (time-respecting) walks. It is found that distinct centrality metrics perform the best in estimating nodal influence depending on the infection probability of the spreading process. For a broad range of the infection probability, a node tends to be influential if it can reach many distinct nodes via time-respecting walks and if these nodes can be reached early in time. We find and explain why the proposed metrics generally outperform classic centrality metrics derived from both full and partial temporal networks.