Will the miner simply ignore that block?
Yes, the block will be just ignored by the miner.
how will the miner update to A-B-C-D-E (step by step)?
When the miner's node gets online and start connecting to peers, it will start a “handshake” by transmitting a
version message, which contains all basic identifying information, including
BestHeight (the blockchain height of the node). Since v0.10, the nodes sync with 'headers-first'. This means that the nodes when restarted first ask peers for block headers and validate those. The
getheaders message requests a headers message that provides block headers starting from a particular point in the blockchain. The connected peers will respond with a
headers message that sends block headers to the node which requested certain headers with a
getheaders message. The disconnected miner's node will then verify the difficulty until the tip. This allows a peer which has been disconnected to get the headers it hasn’t seen yet.
In a second stage, when the headers have been discovered, the node can then download the blocks. However, as the node already know about the whole chain in advance due to the block headers it has received, the blocks can be downloaded in parallel from all available peers. These blocks are received through
The above process describes the syncing of node when it was disconnected for some time. To make the answer more complete, I'll expand a bit further as to how new transactions and blocks are broadcasted. Nodes unsolicitedly transmits
inv message (inventory message) containing one or more inventories of objects known to the transmitting peer. It can be sent to announce new transactions or blocks. The receiving peer can then compare the inventories from an
inv message against the inventories it has already seen, and then use a follow-up
getdata message to request unseen objects.