Running Tezos node with rolling mode but sync started at the beginning

Im not sure that OP is importing a snapshot - seems OP simply wants to run a rolling node, and expects the node to sync itself with only the 5 latest cycles?

Use the history-mode argument when importing the snapshot. These are the steps:

rm -rf .tezos-node/{context,store,version.json}
tezos-node snapshot import "${SNAPSHOT_FILENAME}" --history-mode=experimental-rolling
tezos-node config init
tezos-node run --history-mode=experimental-rolling --rpc-addr=127.0.0.1:8732 --connections=20 --bootstrap-threshold=3