In real-world systems, particularly in neuronal networks, delays are inevitable, yet often overlooked in computational studies, including those exploring synchronization dynamics. Given the critical role of synchronization in brain networks, this study offers an investigation into the impact of delay on synchronization patterns in a duplex neuronal network composed of memristive Rulkov neuronal models. It uniquely examines the combined effects of both short-range (intralayer) and long-range (interlayer) delays on synchronization, providing a more comprehensive understanding of how temporal mismatches influence network coherence. The mathematical model incorporates chemical synaptic functions to represent long-range connections between neurons in different layers. For intralayer communication, two scenarios are considered: (1) electrical synapses link the neurons, and (2) chemical synapses mediate their communication. Synchronization error is employed as a key metric to evaluate the network's coherence. The findings reveal that introducing delay to electrical synapses disrupts interlayer synchronization and reduces the range of intralayer synchrony. Conversely, adding delay to chemical interlayer synapses primarily alters the boundaries between synchronous and asynchronous regions. When both short-range and long-range connections are chemical, the synchronization error exhibits similar intra- and interlayer patterns, with delays in long-range synapses causing more pronounced changes. Notably, this study identifies the emergence of repetitive patterns in the synchronization error, offering new insights into how delays shape the stability and transitions between synchronization states. These results underscore the critical influence of delay in modulating synchronization dynamics, with potential implications for understanding temporal disruptions in neuronal networks.