How to pass value to XSLT stylesheet during run time?

You have to use parameters.

XSL stylesheet

<xsl:stylesheet
  version="1.0"
  xmlns:xsl="http://www.w3.org/1999/XSL/Transform">

<xsl:param name="date">DEFAULT-DATE</xsl:param>

...

Calling via Ant

<xslt in="${xml}" out="${out}" style="${xsl}">
    <param name="date" expression="${DATE}"/>
</xslt>

javax.xml.transform.Transformer.setParameter(name, val);