A few thoughts:
First, Frank Harrell (who developed the C-index) doesn't think that it is useful for comparing models. See this answer and its links, especially to a web post by Harrell. The C-index nicely summarizes how well a single model discriminates between individuals, but it says nothing directly about the calibration of the model: how well the model's predicted survival estimates correspond with what was observed. The link contains references to better ways to compare models.
In response to edited question: The Antolini index, as I understand it, is still just a measure of discrimination. It evidently extends the C-index to a situation with time dependence. It does not seem to provide any more information about calibration than the C-index.
If you are interested in the quality of a prediction "for a single sample," as you say, then what you need is a measure of calibration instead. That's typically done by estimating "observed" probabilities of survival at some time by a very flexible fit to the data set and comparing against model-predicted probabilities. Repeating the process on resampled data sets can give corrections for optimism in the fit. See this page for an outline.
The potential problem with a calibration measure is that your use of the Antolini index suggests that you have time-dependent covariates in your model. In that situation I don't know of a reliable way to estimate "observed" and "expected" probabilities of events at any given survival time, at least in a way that can extend reliably to new data samples.
The problem (at least for survival models with at most one event per individual) is that if you have a covariate value for an individual at some time, you already know that the individual is alive at that time. The
lifelines package, for example, thus won't even allow for predictions from Cox models with time-dependent covariates. Harrell's
calibrate() function in his R
rms package won't handle them, either. There might be ways around that with a joint model of covariates and survival over time, but that's beyond my expertise.
Second, unless you have tens of thousands of cases, using separate training and test sets isn't a good idea. See this post by Frank Harrell.
In response to edited question: If you have a large enough sample to set aside a completely separate test set, then you already can compare your discrimination measure directly between the two models. The Wilcoxon-Mann-Whitney test is just a discrimination test, a rescaling of the C-index. See the section of this answer about that test.
Resampling via cross-validation or bootstrapping is close to what you suggest for getting estimates of variability; explaining what you did to others would be easiest if you directly invoked one of those approaches. They have the advantage that you could compare the whole modeling process between your two training methods. For example, you could repeat each of the training methods on multiple bootstrap samples of the training data and evaluate the discrimination either on the entire data set (for a small data set) or on the held-out test set (for very large data set). That would compare the performance of the two training methods directly.
Third, an answer would depend on what you mean by the "baseline model." Trivially, if the baseline model is a null model (no model at all, no predictors) and you really wanted to use the C-index, then the standard error reported for the C-index (concordance) would indicate whether it's statistically different from the value of 0.5 expected from a null model.
If the "baseline model" is a model only using a subset of the predictors in your complete model (that is, the baseline model is nested within the complete model), then a likelihood-ratio test between those models would be a better-accepted and more sensitive comparison. (Again trivially, the likelihood-ratio test reported just for the complete model also documents its superiority to a null model.)
In response to edited question: These don't seem to be nested models of the type that a likelihood-ratio test could evaluate.