For example, it's a Fleecemane Lion. I pay mana to make it monstrous. Can the opponent fatal push it in response to get rid of it before it can become indestructible?
Yes. Fleecemane Lion has a normal activated ability, which means that a player can respond to it while it’s on the stack and before it resolves. So, your opponent can cast Fatal Push in response to the Lion’s ability being activated, which results in the Lion being destroyed before it becomes monstrous or gains a +1/+1 counter (and therefore before it becomes indestructible).
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Want to play a UW control deck in modern, but don't have jace or snaps?
Please come visit us at the Emeria Titan control thread
Commander/EDH Decks:
BRG The Blood of Jund - Kresh the Bloodbraided BRG
WR The Blades of Goldnight - Gisela, Blade of Goldnight WR
Nexus MTG News // Nexus - Magic Art Gallery // MTG Dual Land Color Ratios Analyzer // MTG Card Drawing Odds Calculator
Want to play a UW control deck in modern, but don't have jace or snaps?
Please come visit us at the Emeria Titan control thread
RULES OF MAGIC :
http://magic.wizards.com/en/game-info/gameplay/rules-and-formats/rules
Nexus MTG News // Nexus - Magic Art Gallery // MTG Dual Land Color Ratios Analyzer // MTG Card Drawing Odds Calculator
Want to play a UW control deck in modern, but don't have jace or snaps?
Please come visit us at the Emeria Titan control thread