Wow, what a pairing for a quarter-final. Both players have played in finals. Shirov lost against Anand in 2000, Adams against Kasimdzhanov in 2004. And Shirov of course famously beat Kramnik in a supposed candidates final 1998, only to see Kramnik play the world championship match two years later. But this is just to show that both of the players have a lot of experince in matches, not to start talking about chess politics here.
Shirov - Adams
Shirov plays 1.e4 most of the time, but also has a great deal of experience with 1.d4.
Against 1.e4 Adams plays the closed Ruy Lopez, which Shirov allows with White. Adams most of the time plays the Marshall, which Shirov allows only sometimes.
As I said in my preview of the Carlsen-Aronian match, I am not convinced of the various Anti-Marshalls, at least from a theoretical point of view. On top of that, Adams also sometimes plays the Petroff, which may just be what he is going to do in a match.
Therefore I think Shirov - unless he has some excellent preparation against the Marshall proper - should actually look at playing 1.d4. Here Adams likes to play 1.d4 Nf6 2.c4 e6 3.Nc3 Bb4 4.Qc2 O-O 5.a3 Bxc3 6.Qxc3 Ne4, which looks a bit cheesy to me. In NIC yearbook 62, Adams is quoted: "It isn't very good, but my results are reasonable and I keep playing it". Maybe Shirov can prepare something here.
Adams is a 1.e4 player and almost never plays anything else.
Against 1.e4 Shirov in his career has probably tried almost every decent opening, and even some not so decent ones. In recent years he has limited himself to various closed Spanish systems, a few Petroffs, some Najdorfs, and a lot of Sveshnikov Sicilians.
Adams seems to allow the Petroffs and the Najdorfs. He goes for the closed Spanish unless it is a Marshall, but never allows the Sveshnikovs recently. Instead of a Sveshnikov Adams plays 1.e4 c5 2.Nf3 Nc6 3.Nc3 or 3.Bb5.
I think Shirov should play the Marshall move order of the closed Spanish with Black. Adams then has to either play some Anti-Marshall, where he hasn't won a single game recently, or show his hand against the Marshall proper, which he probably intends to use in this match himself with Black.
This match promises to be very exciting - I only hope we don't see too many boring Anti-Marshalls.