If I play a Hidden Blade at the battlefield I control, on a different turn can I activate it and use it on a unit at a different battlefield?

No, you cannot. When you play a card from "Hidden," you are restricted to targeting units at the battlefield where that card was hidden.

According to Rule 737.1.d.2:

"If a hidden spell or a play effect of a hidden permanent chooses any targets, those targets must be chosen from among options at that battlefield, unless the ability explicitly restricts targeting in a way that makes this impossible."

Additionally, the FAQ for Hidden Blade explicitly clarifies this interaction:

• Ruling: "Hidden Blade can only be used on the battlefield where it is hidden. It cannot target units at other battlefields."
• Nuance: This restriction applies even when the ability is activated on a different turn.

If you wish to target a unit at a different battlefield, you must play the card from your hand for its normal cost, rather than playing it from "Hidden."