No, this is not possible. As Lupus said the rift will bounce the drover and any tokens it would create. This is all done simultaneously - there is never a point where some of the creatures are bounced but others still exist on the field. Drover will still trigger, but nothing will happen until after rift has finished resolving and Drover is no longer on the field. The relevant rules on triggered abilities are here from the Comprehensive Rules:
603.2 Whenever a game event or game state matches a triggered ability’s trigger event, that ability automatically triggers. The ability doesn’t do anything at this point.
The game notes that the conditions have been met, but nothing actually happens, the triggers don't go on stack yet, and the spell continues to resolve.
603.3 Once an ability has triggered, its controller puts it on the stack as an object that’s not a card the next time a player would receive priority. See rule 117, “Timing and Priority.” The ability becomes the topmost object on the stack. It has the text of the ability that created it, and no other characteristics. It remains on the stack until it’s countered, it resolves, a rule causes it to be removed from the stack, or an effect moves it elsewhere.
So what happens is the triggers are met while rift is resolving, then rift finishes resolving and the triggers are put on stack, this is the time you can respond to the triggers going on stack, they start to try to resolve, and that too can be responded to, but since Drover is now back in the hand, the abilities do nothing. At no time does any of the triggers caused by and overloaded cyclonic rift or similar spell (any mass wipe or bounce) actually put any counters on the Drover.
You could use drover's ability to remove counters that are already on it in response to Cyclonic Rift, but unless you have something that counts creatures entering or leaving the battlefield, that won't change anything other than costing you mana - the tokens will be bounced at the same time as drover and since they are token they will cease to exist.