You just need to adjust the font size with Table.set_font_size (after turning off the auto-size) :
the_table.auto_set_font_size(False)
the_table.set_fontsize(12) # <- adjust the size here
And for the borders, you can make thin borders with a table.Cell.set_linewidth for each cell separately:
cell.set_linewidth(0.3) # <- adjust the width here
The full code :
def xlsx_to_pdf(filename):
    wb = openpyxl.load_workbook(filename)
    for ws in wb:
        df = pd.DataFrame(ws.values)
        with PdfPages(f"{ws.title}.pdf") as pdf:
            fig, ax = plt.subplots()
            ax.axis("tight")
            ax.axis("off")
            the_table = ax.table(cellText=df.values,
                                  colLabels=df.columns,
                                  cellLoc="center",
                                  loc="center")
            the_table.auto_set_font_size(False)
            the_table.set_fontsize(10)
            for _, cell in the_table._cells.items():
                cell.set_linewidth(0.5)
            pdf.savefig(fig, bbox_inches="tight")
xlsx_to_pdf("file.xlsx")
Output :
