can arduino UNO be programmed to measure the analog voltages of 6 resistor ladder network without using switches
Yes, it can. You can connect the 3.3V as the external ADC reference and use 5 of the 6 available ADC input pins to measure the voltages at each node. 

The AREF pin has relatively low impedance and you need to make sure that the 3.3V is not applied when the 5VDC supply to the Arduino is less than 3V. A Schottky diode from AREF to the 5V supply may be necessary. 

Alternatively, you could use the default 5V reference and measure the 3.3V as well, which requires 6 analog inputs. That's perhaps easier but less accurate. 

In either case you need to ensure that voltages applied to the analog input pins does not exceed the supply voltage to the ATMega chip (even if the 5V supply is off) or if it does, that there is sufficient resistance in series (such as a few K ohms) to limit the current.

As to the code required, that's up to you.