Is it possible for someone to get an address the old way today? How long would it take?
If you are mining I suppose you have a certain amount of money. Having said that I assume you have a bit of other money staying with you too. Use this to buy a hardware wallet to keep your earnings from mining there. It is in my opinion the safest bet for you and the very best security for storing your bitcoins.
How long it would take ? Well depending on your country after you order it can take 5-15 days as a maximum (business days).
A few wallets are Trezor, Ledger, Keepkey. The cheapest is Ledger so you can google Ledger wallet to buy one for yourself.