Can you play units from base to a battlefield during a showdown if they are ready?

No, you cannot use a standard move to move units from your base to a battlefield during a showdown.

According to Rule 144.1.c, a standard move "cannot be performed during a Showdown or Combat."

As noted in the RiftJudge FAQ (ID 2229):

• You cannot use standard moves to move units to a battlefield once a showdown has started.
• You can only play or move units to a battlefield during a showdown if an effect specifically allows it (such as a spell that moves units, or a unit with an ability that allows it to be played during a showdown, like Shen).

Whether the unit is ready or exhausted does not change this restriction, as the standard move action itself is prohibited during a showdown.