Yes. You can choose a Pestilence on the battlefield and any future activations will have their damage prevented for that turn for the targetted creature or player.
Note that you target both a creature, player, or planeswalker and a card in an opponent's graveyard when you activate Shieldmage Advocate's ability (C.R. 115.4, 108.1), but you don't choose a source of damage until that ability resolves (the source of damage isn't a target because it's not identified with the word "target" [C.R. 115.1c; see also C.R. 115.10a]); therefore, other players won't know what that source will be before then.
EDIT (Apr. 28, 2018): Edited to conform to changes with Dominaria.
EDIT (Aug. 16, 2018): Correctness edit.
EDIT (Feb. 18, 2021): Some rules were renumbered in the meantime. Add rule citation.
Does this card prevent future damage from permanent cards, like Pestilence.
[Inkfox Aesthetics] [The Spirit of EDH] U Sakashima the Impostor R Kiki-Jiki, Mirror Breaker Sigarda, Host of Herons
EDIT (Apr. 28, 2018): Edited to conform to changes with Dominaria.
EDIT (Aug. 16, 2018): Correctness edit.
EDIT (Feb. 18, 2021): Some rules were renumbered in the meantime. Add rule citation.