Young man here's a little advice from someone who has been romancing the same young lady for 38 years.
For Christmas I would go with jewelry you don't have to be extravagant just something pretty like earrings or one of those heart necklaces where you keep half and they have the other you want something that will remind her of you.
For the birthday I recommend flowers while roses are always first rate if you are on a budget you can find any number of less expensive bouquets.If you can have them delivered to her at her place of work that's even better.Follow that up by taking her somewhere quiet and cozy for a candle lit dinner.Again the intimacy and ambiance of the restaurant are more important than the price.Making her a candle lit dinner is another option if you are handy in the kitchen.
'I will live in the Past, the Present, and the Future!' Scrooge repeated, as he scrambled out of bed. 'The Spirits of all Three shall strive within me. Oh, Jacob Marley! Heaven, and the Christmas Time be praised for this! I say it on my knees, old Jacob, on my knees!' (Charles Dickens A Christmas Carol)