Keep yourself posted on flight comparison sites regularly to snap up cheap first class flights. The fact is flight prices tend to fluctuate every month, so the best thing would be to check the airline sites regularly for the best possible deals. You can also set up price alerts with different websites so that you can regularly receive emails informing you about the change in fare for the flight you want to book.
There are also multiple apps that you can make good use of on occasions like this. Through these apps, one can come to know about a suitable time to purchase the ticket of your choice.
Subscribe to the email newsletter to get alerts on first-class fares. You can get useful information on last minute flight deals and other info on airfares.