I have been trying to figure this out for a week now.
Yes in custom blueprint for the object you can set that in event tick.
Or in the level blueprint make it move in the event tick or with a looping timer 
Is there a way to make the object move up and down automatically as level begins?
Do you already try to use Timeline ? It´s quite usefull, and easy to implement direct in your BP :
well iam too stupid to understand it. When i set some Z values it rotates or goes up but there is no loop in up and down.
sorry, to bother you.
using a set node is a one time thing if you want it to loop then you need to add more script to make that happen. for example you could use a timeline and update the set node many times over a given period. or you could set offset on tick which would update the location every frame. then you would need to create some logic if you wanted it to change directions such as comparing the location to a predetermined location or limiting the number of time that the set location can be done before you change the delta to a negative or just running the timeline in reverse once its completed.
You could also make a simple matinee with an event track if this is something that persists in the level from the beginning.
Don’t know if this is still relevant but I was struggling with this recently.
I’ve discovered probably the easiest method for simple local up and down translation. Create a timeline in the actor’s blueprint that looks like an amplitude. Like this:
Then, I used a AddRelativeLocation node. Split the transform input pin to expose the float inputs, then just input the float track output from your timeline in your desired channel. I used the Z transform in my example:
Result: