Does the RB5009UG support EEE (Energy-Efficient Ethernet)? If supported, Does it support EEE for copper ports or fiber port, or both?
I have checked the specifications of RB5009UG, but couldn't find it.

RB5009 does support EEE 802.3az on ether1 - ether8.
I have several RB5009s connected to Cisco switches and EEE is supported and operational on Cisco <-> RB5009 1GB connections.
It is not mentioned in the MT specs, It is not visible, cannot be disabled. But its there and it works.

Here an example of a Cisco SG350 switch connected gi1 -> RB5009/ether3:
EEE is supported and active.

sg350-paca01# show eee gi1
Port Status: up
EEE capabilities:
Speed 10M: EEE not supported
Speed 100M: EEE supported
Speed 1G: EEE supported
Speed 2.5G: EEE not supported
Speed 5G: EEE not supported
Speed 10G: EEE not supported
Current port speed: 1000 Mbps
EEE Remote status:enabled
EEE Administrate status: enabled
EEE Operational status:enabled
EEE LLDP高级strate status: enabled
EEE LLDP Operational status: disabled
Resolved Tx Timer: 17
Local Tx Timer: 17
Resolved Rx Timer: 17
Local Rx Timer: 17