Regularization with LASSO, ridge, or elastic net is suitable for even $n<<p$ scenarios, so from the information provided in the question alone, there is no apparent reason to use another method beforehand to pre-select variables.$^\dagger$
However, if you are concerned that there are a lot of non-influential variables, you might consider using LASSO twice,$\ddagger$ re-estimating a subset of the parameters after an initial selection of which are non-zero, which is not an uncommon approach.
In fact, using just elastic net or double LASSO might be a better approach than first selecting variables using some other method, because this may cause the coefficients of the selected variables to be positively biased. Although it focuses more on inferential statistics than classification, you may want to read this related answer, specifically the section:
[A]ll predictors in a model and their posited causal relationship between a single exposure of interest and a single outcome of interest should be specified apriori. Throwing in and excluding covariates based on their relationship between a set of main findings is actually inducing a special case of 'Munchausen's statistical grid' (Martin, 1984).
$^\dagger$ Zou & Hastie (2005): Regularization and variable selection via the
$^\ddagger$ Meinshausen (2006): Relaxed Lasso