To know what is happening at the other end of its body,
This creature passes through time the same way a fence passes through a field. The fence has a start and an end and everything in between, but in reality the fence is there all at once. Similarly the creature already exists at all points in time at once.
Imagine you are lying down in a field, with your right hand being in the past and your left hand being in the future. There are some special ants that live according to field time. They are born at the right side of the field and due to some unexplained force move leftwards throughout their lives. They interpret field time as real time.
One day an ant bites your right hand and by instinct you close your left hand. This crushes the ants walking on your left hand. From the ants' point of view you are a high-dimensional creature that has remembered the bite in the past and used it to act in the future. From your point of view however it all happens at once. It's not a memory but just your sensory system.
The creature works the same way. What you might call memories it just calls its nervous system.
Of course none of this answers the question of how a full-dimensional creature would experience its lifetime if its consciousness does not pass through a timelike dimension. In the field example there is the higher dimensional real time and your consciousness passes through it linearly. You can act in the ant's past but not in your own past.
How does a five-dimensional creature work if there is no sixth dimension for its consciousness to move along? It throws up loads of other questions. You also need rules for avoiding time paradoxes.*
*Assuming you're into that sort of thing. It's your world after all. Feel free to have as many paradoxes as you want. I promise I won't tell mum.