It depends on how security conscious you are. Technically you can route things through remote nodes and thus avoid downloading the blockchain. But the monero co

It depends on how security conscious you are. Technically you can route things through remote nodes and thus avoid downloading the blockchain. But the monero community is security conscious and usually recommends downloading the blockchain, which takes a while if you do it the “right” way and don’t just find a copy of it hosted somewhere and download it. If you are ok with skipping that, you can use something like CakeWallet to create a wallet on your phone and then give someone a receiving address