This is how they calculated:
Molar mass of lithium: M=6,941 g/mol
Density of solid lithium: 515 kg/m^3 = 515000 g/m^3 = 74196,8 mol/m^3 = 1,3477*10^-5 m^3/mol --> this is the volume of 1 mol lithium
1,57% volume change: (1,3477*10^-5)*1,0157 = 1,3689*10^-5 m^3
dV=(1,3689-1,3477)*10^-5 = 2,12*10^-7 m^3
Now, we have this equation here:
dT/dp = (Tm*dV)/dfH
Where dT is the change of the melting point in Kelvin, dp is the change of pressure in Pascal, Tm is the standard melting point, dV is the change of volume in m^3, and dfH is the enthalpy of fusion in Joule/mol.
So you have every data you need to find dT.
dp=1,01223*10^8 Pa
Tm=453,69 K
dV= 2,12*10^-7 m^3
dfH= 3024 Joule/mol
So dT=3,2195 K , which means that the new melting point (at 1000 atm) is 456,9 K