1) The answer depends on whether the plan year ends in 2023 before or after your hire date. If the employer plan uses a calendar year, you will be considered to have been covered by the employer's retirement plan for the entire year. Your question suggests that you are married and will be filing jointly with a MAGI of at least $136,000, so all of your traditional IRA contributions for 2023 will be nondeductible when covered by a workplace retirement plan for 2023. If your spouse is not covered by a workplace retirement plan, the MAGI limit to be able to deduct a full annual traditional IRA contribution for your spouse will be $218,000 and will phase out entirely are $228,000.
If the plan uses a fiscal year that has an plan-year end date in 2023 prior to your hire date, you won't be covered for 2023 (unless you make self-employed retirement contributions for 2023). You are considered covered for the year if there are additions to your account for the plan year that ends with or within that year.
2) Even being covered by a workplace retirement plan for only one day for the year makes you covered for the entire year. Your 2023 Form W-2 will have box 13 Retirement plan marked to indicate that you are covered for 2023 if you are indeed covered for 2023.
If your MAGI is under $218,000 and you file jointly, you might consider recharacterizing your traditional IRA contributions to be Roth IRA contributions instead.
