Node A is holding a chain which is not in sync with the rest of the
nodes and node A decides to post a transaction.
If node A created transaction from valid unspent UTXO (which is valid on longest chain), the transaction will be valid. You don’t have to be even a node to send this valid, signed transaction, you can mail it to the node C which will include it into mempool.
The miners will pick up this transaction and add them to a block in
their own chain which is different than Node A’s chain.
As long as your node has communication with these miners, yes (there are different scenarios of “unsyncing”, I assume hardforks as well, but I guessed you meant orphan blocks mostly, or lack of recent blocks?).
So now when Node A looks at the network, will it just ignore its own
inconsistent chain and pick the miners chain?
No. If you are looking at top of chain A which isn’t in sync with the longest chain, where miners should add the transaction, your node won’t see it. The rest of the network on the longest chain will observe it though.
Will the transaction made by node A even show up in the blockchain?
Yes, in the longest chain.