In the reinforcement learning (RL) tasks, the ability to predict receiving reward in the near or more distant future means the ability to evaluate the current state as more or less close to the target state (labelled by the reward signal). In the present work, we utilize a spiking neural network (SNN) to predict time to the next target event (reward - in case of RL). In the context of SNNs, events are represented as spikes emitted by network neurons or input nodes. It is assumed that target events are indicated by spikes emitted by a special network input node. Using description of the current state encoded in the form of spikes from the other input nodes, the network should predict approximate time of the next target event. This research paper presents a novel approach to learning the corresponding predictive model by an SNN consisting of leaky integrate-and-fire (LIF) neurons. The proposed method leverages specially designed local synaptic plasticity rules and a novel columnar-layered SNN architecture. Similar to our previous works, this study places a strong emphasis on the hardware-friendliness of the proposed models, ensuring their efficient implementation on modern and future neuroprocessors. The approach proposed was tested on a simple reward prediction task in the context of one of the RL benchmark ATARI games, ping-pong. It was demonstrated that the SNN described in this paper gives superior prediction accuracy in comparison with precise machine learning techniques, such as decision tree algorithms and convolutional neural networks.