ESP8266 with Arduino Uno and HTTP POST
Yes, the library 


https://github.com/bportaluri/WiFiEsp


Is suited for your task because it implements talking to a ESP8266 over a software serial. On this serial connection, AT+ commands will be sent. This also means that you must have the AT firmware flashed onto your ESP-01.

If that is not the case you can download the binary image here and flash it with a serial adapter, as instructed with the tools on that page. You can check whether your ESP8266 has the firmware flashed by attach a UART adpater to the TX output. It should print a "ready" on bootup.

For your wire-up, you will just have to add VCC (+3.3V), GND, TX and RX connections to the ESP-01 board, preferable with a big capacitor between VCC and GND, as the ESP8266 draws a lot of power when sending data over WiFi.