The method calls for 20 mM HEPES pH 7.2. You can prepare this in a number of ways. One typical way is prepare a stock solution of 0.5 or 1 M HEPES from solid, with the pH adjusted to 7.2 using NaOH. This is then diluted to make the 20 mM buffer.
The whole buffer composition is 20 mM HEPES buffer, pH 7.2, 1 mM EGTA, 210 mM mannitol, and 70 mM sucrose. Start from concentrated stock solutions. You can buy a 0.5 M stock solution of EGTA (wear gloves when handling this solution). Make 1 M stock solutions of mannitol and sucrose. Filter sterilize all of the stock solutions and keep them in the refrigerator.
If it is sterile, it should last for a long time. However, with the carbon sources in it, if it is not sterile it will grow bacteria and mold, and I would not expect it to last more than a few days. The concentrated stock solutions, in contrast, can be stored for months because of their high osmolarity (although an effort should be made to keep them sterile), so you could make up a fresh buffer each day, as needed.