A market order or a limit order can be placed. Rather than placing a market order, I usually place a limit order and wait for the price to come to me.
Price may need time to settle depending on the option you are trading due to initial volatility early in the session, or after a large quick move, etc. Unless they are a few days from expiry and gamma moves the delta up faster, I have seen the price of calls not increase after the underlying has moved up quickly.
If you have a limited order, you can watch and monitor price to get a sense of where the option's price should be once the underlying's price reaches the level at which you want to buy the option. It's a type of price discovery that I frequently employ.

Yes, you can place after-market orders for options, but it depends on the brokerage platform you are using. After-hours trading allows investors to place orders outside of the regular trading session, typically before the market opens or after it closes. However, option contracts are not as commonly traded in after-hours markets as stocks.

Some brokers may allow you to place orders that will be executed when the market opens, but liquidity is usually much lower, and spreads can be wider, increasing potential risks. It's important to check your broker’s policies regarding after-market orders for options and consider market volatility, as prices can fluctuate significantly between the close of one session and the opening of the next.