How long does it take to replace a battery of MacBook Air?

You can see the cost by checking on this Apple support page here:

Mac Repair and Service - Apple Support

As for how long it will take for Apple or an AASP to replace the battery depends on many factors (region, part availability, mail-in repair versus repair being done at service location, and whether any other issues may come up during the repair, and how busy the service location is). Generally consider it will take about a week. Some may be repaired in as little as two days if everything lines up while others could take a week....sometimes longer. You can ask the service location (Apple Store or AASP) for the general turn around time, but keep in mind it is just a guess since they are at the mercy of Apple for shipping the part, the delivery service shipping the part for timely deliveries, etc. Many places may not even provide you with an ETA because they know some people will take any "guess" as 100% guarantee......there can be no guarantees on turn around time.

It depends on where you are livibg. If in e.g. Belgium or the Netherlands and if you can make an appointment in an Apple Service Centre, it'll take a couple of hours and it costs about 185 euros