Not with any certainty, no. The designations don't come until after rollover into the new season.
The type A/B designations are based on the player's overall rating and the position that the AI would consider his primary position. So for instance, if he has shortstop ratings but you play him in LF, he would still be grouped with other shortstops.
With experience, you'll get a feel for how the player is likely to be designated (or not). In the meantime, your best bet would be to look at the current season free agents to see who the A's and B's were, and try to find the closest match to your guy. You can find them on the "Draft Order" page. Won't be perfect, but it will be a rough approximation.