Curtis Granderson is out until May. Mark Teixeira is out until May. The Yankees had enough depth to make up for the loss of an outfielder, but now an outfielder and a first baseman? That's going to be tougher. This is also the most difficult time to fill a need because teams are still evaluating what they have and who they don't need. If it was earlier in the offseason there might have been more free agents available to sign, if it was later then teams would be more inclined to trade off the spare parts they have no use for.
In all likelihood the Yankees will use some combination of Juan Rivera, Dan Johnson, Kevin Youkilis and Jayson Nix to fill first base, keep third base strong and keep someone in the outfield. Three of those players on the same team is kind of disgusting, but it will get the team through the first month of the season. The Yankees could also use Corban Joseph at third, but he hasn't been incredibly impressive (4 errors in 6 games) in Spring Training so far.
If the Yankees want to find an upgrade that won't cost them their coveted prospects they could peek their collective heads into other teams' Spring Training camp. There they will find spare parts that could hold the fort until this whole ordeal is over and everything is back to normal.
Most importantly, if the Yankees want to look for outside options they actually have to be upgrades over what is already in camp. Dan Johnson might not make people feel confident, but in his limited time in the majors he's put up a 102 wRC+ and has a pretty even 103/102 L/R platoon split. He is still probably an upgrade over the likes of Kila Ka'aihue, Jeff Baker, Matt LaPorta and Casey Kotchman. Jeff Clement has a 102 wRC+ against lefties, but he would need to be platooned against righties. Brad Hawpe has the reverse problem because he has a 119 wRC+ against right handers, but is below average against left handers.
If they really dislike Johnson they could trade for both and platoon them together and keep Youkilis at third, but that's certainly not happening. It's not worth the effort. Johnson will make a fine replacement for the time being. They could acquire Hawpe to be the starting first baseman and have Johnson off the bench to hit against lefties, but still, what's the point?
The pickings at third base are even slimmer. It's not very difficult to be better than Jayson Nix. He has a 94/62 L/R platoon split and a -1.1 UZR/150 and yet that doesn't leave many options. Donnie Murphy, Brandon Inge and Kevin Kouzmanoff could all be useful to the team, but somehow Nix has a better bat than all of them. The only upgrade any of them could bring is on the field.
Murphy currently has a -3.3 UZR/150 at third base, but that's due to a bizarre 2012 where he put up a -37.8 in 152.1 innings. The last time he had more than 100 innings in a season at third he had an 11.1 and has never put up negative value at third base until last season. He also comes with the added bonus of being able to play all over the infield. Brandon Inge and Kevin Kousmanoff are not very good hitters, but they do have good gloves. Inge has a 6.8 UZR/150, while Kouzmanoff has a 5.5 UZR/150 at third base and they could offer a very good fielding alternative to Nix. However, would these players really make the team better? With Alex Rodriguez out and Kevin Youkilis playing first base in this scenario, the team needs more offense, not less, at third base.
The Yankees could certainly make any of these moves if they want another body without having to give up much, unfortunately, these alternatives are mostly downgrades to what we have now. Unless the Yankees want to make a more significant trade, which they probably shouldn't, considering both Granderson and Tex will be back, they should do what they can with the pieces they already have. It's not pretty, but that's life.