Rodriguez was examined in New York on Monday by team hand specialist Melvin Rosenwasser, who diagnosed the sprain and administered the shot.
"They told me basically to stay off it, to pretty much do lower-body stuff, stay away from the bat for a day or two," Rodriguez said.
"The one thing is it's so frustrating to come here. It's the last time we come here for the regular season. It hurts just sitting on the sidelines, just watching."
Rodriguez originally hurt the thumb backhanding a Joe Mauer infield single on Aug. 21 at Target Field and aggravated it in the first game of a day-night doubleheader against the Orioles on Sunday.
The injury is the third notable one involving Rodriguez this year. He also had surgery to repair a torn meniscus in his right knee and has been battling soreness in his left shoulder.
"This year's been the most frustrating of my career because I've had three different things going on at some point or another," Rodriguez said.
"But my body feels really good. And when I'm healthy, I feel like I can do things that I did in April and in Spring Training. It's probably as good of baseball as I've played in a long time. If my health is there, I can play through pretty much anything, so long as I don't re-injure it."
Rodriguez said that the thumb felt "a little sore" after taking ground balls at third base off the bat of infield coach Mick Kelleher. Rodriguez had his thumb wrapped in the clubhouse before batting practice.
Yankees manager Joe Girardi said that he potentially could use Rodriguez as a pinch-runner or to play defense in the Red Sox series, but it is unlikely he will play a major role.
"Would I be surprised if he didn't play in this series? I wouldn't be," Girardi said. "Am I hopeful? Yes, that we'll get him tomorrow or maybe the next day. But I wouldn't be surprised if we didn't have him."
Rodriguez said he isn't giving up on playing in the Boston series, but a return on Friday against the Blue Jays at Yankee Stadium is more probable.
"It's miserable," Rodriguez said. "It's painful to come here and you have just a perfect day to play baseball at Fenway Park. It's one of my favorite places to play."