Something like this would work, though you would have to adjust rho by trial and error to get the final corr(x1,x2) value that you want:
% Just example values for some parameters--I was not sure what values you
% want.
alpha = 1.1;
beta = 4.2;
Ast_mean = 23.45;
Ast_std = 6.789;
LB = 41.368542; %lower bound
D = 1337.582858; %difference between upper bound and lower bound
nsam = 100000; % generate a large sample so we can estimate the true correlation accurately
rho = 0.5; % adjust this up or down to get the desired corr(x1,x2)=0.5 (or close enough)
% You shouldn't have to change anything from here
x = mvnrnd([0 0],[1 rho; rho 1],nsam);
xcdf = normcdf(x);
x1_aux = betainv(xcdf(:,1),alpha,beta);
x1 = x1_aux*D+LB;
x2 = x(:,2)*Ast_std + Ast_mean;
corr(x1,x2)
