Predicting class probabilities in regression based on area under the curve Logistic regression models the log odds. That is for rv $Y$ which is binary 
logit$(Y=1)=X\beta$. 
Then with this model, you can estimate the class probabilities and hence prediction or classification is immediate. Ordinal regression does a similar thing with cumulative logits with which the class probabilities can then be calculated. Now looking at $AUC$ regression where Dodd and Peppe(2003) modeled the $AUC$, by 
$AUC$=$P(Y<Y^{\prime}$ $|$ $X,X^{\prime}$)=$f(X\beta)$ where $X$ are covariates. My question is how do we now do predictions with such a model. 
 A: Logistic regression models the probability of a binary response. We use some sigmoidal function to predict a value that lies between 0 and 1.
The logit function is just one popular link function. Why? It goes between zero and one. Probit is another link function. $\Phi(X\beta)$ uses the cumulative distribution function of a normal to do the same job as logit.
AUC is another function(al?) that goes between zero and one. You could also look at it as a variable between 0 and 1 that is to be predicted. Having had a look at the paper you mentioned I think they are using logistic regression to model the AUC, e.g. from the following quote:

When the logit link is used, exponentiated parameters have interpretations as AUC odds, where AUC odds are defined as [...]

A: You make predictions in exactly the same way as in the ordinary model. AUC is a summary of how well the model separates the two classes (similar to how $R^2$ is a summary of the model's improvement in explained variance compared to the mean of the response), but has no bearing on the task of predicting class membership $\hat{y}$ given some covariates $X$.
