When a showdown starts, is my opponent able to play a ready card from his base to the battlefield when it's his priority?

No, your opponent cannot play a ready unit from their base to the battlefield once a showdown has started.

According to Rule 144.1.c (as referenced in RiftJudge FAQ ID 9703), a standard move "cannot be performed during a Showdown or Combat."

Furthermore, as noted in the FAQ:

• 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).

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