What happens when a node goes offline for a while?
Full nodes are always in a perpetual state of "catching up to the network".  This is true if they are just booting up for the first time, have been down for a month, or have been running since the network began.  Nothing is fundamentally different about their operation.

The amount of time a node will take to catch up is determined mostly by a) how many blocks/transactions it still needs to verify, and b) how fast the node is at verifying blocks/transactions.  To a lesser extent, it is also determined by network speed (how fast it can receive blocks/transactions), but the bottleneck is usually the computational power of the node.